Research Article Involutions in the Automorphism Groups of Small Sporadic Simple Groups Chris Bates, Peter Rowley, and Paul Taylor School of Mathematics, e University of Manchester, Alan Turing Building, Oxford Road, Manchester M13 6PL, UK Correspondence should be addressed to Peter Rowley; [email protected] Received 31 July 2014; Accepted 26 November 2014 Academic Editor: Peter Fleischmann Copyright © 2015 Chris Bates et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. For each of fiſteen of the sporadic finite simple groups we determine the suborbits of its automorphism group in its conjugation action upon its involutions. Representatives are obtained as words in standard generators. 1. Introduction Groups permeate many areas of mathematics. Sometimes they have cameo roles; other times they are centre stage. Frequently it is involutions, elements of order two, that are in the spotlight. For instance, in the topological arena we have involutory maps on the 3-sphere in connection with the Smith Conjecture (see, e.g., [1–3]), while in Banach spaces we encounter such things as involutive gradings and fixed points of involutions (see [4, 5]). In areas of algebra, such as associative rings and algebraic groups, there are many sightings of involutions (see [6–8]). Involutions can oſten have a considerable influence on the structure of the group to which they belong. Even their absence can be telling— witness the Feit ompson theorem [9]. For a finite group of even order, Brauer and Fowler [10] establish many results concerning involutions and other properties of the group. For example, they bound the index of a proper normal subgroup in terms of the number of involutions the group possesses. In a similar vein, for a finite group with at least two conjugacy classes of involutions the ompson order formula ([11], eorem 35.1) gives its order using data closely associated with the involutions. In the case when we have a finite nonabelian simple group, more oſten than not, its involutions play a dominant role (see, e.g., [12]). is paper studies the involutions in Aut() where is a small sporadic finite simple group. By small we mean that is isomorphic to one of the following groups: 11 , 12 , 1 , 22 , 2 , 23 , , 3 , 24 , , , , , 3 , 2 . e diminutive appellation aligns, more or less, with said group having a nontrivial permutation representation of degree at most 6156. Several of the larger sporadic groups have been studied individually in [13–16]. So for the remain- der of this paper is assumed to be a small sporadic simple group and is a subgroup of Aut() containing . Also will denote an involution of . Put = , the -conjugacy class of . Our aim is to study the suborbits of in its conjugation action on , or, in other words, to determine the action of () on \ {}. is we do employing the services of the computational algebra packages Gap [17] and Magma [18] partnered by the electronic Atlas [19]. It goes without saying therefore that we use the Atlas notation and conventions as given in [20]. 2. Calculating Orbit Representatives As our starting point we take the smallest nontrivial per- mutation representation of as described in [19] with being generated by standard generators denoted here, as in [19], by and . In the case when ̸ = , the standard generators for are, again as in [19], denoted by and . Having chosen a suitable element in to play the role of , we then forage for elements 1 , 2 ,..., ∈ so that {, 1 ,..., } is a complete set of representatives for the ()-orbits of . In doing this we make frequent use of the standard command IsConjugate(H,x,y)—this works effectively here since the degree of the permutation representation of is no more than 6156. Also observe that Hindawi Publishing Corporation Algebra Volume 2015, Article ID 587629, 15 pages http://dx.doi.org/10.1155/2015/587629
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Research ArticleInvolutions in the Automorphism Groups ofSmall Sporadic Simple Groups
Chris Bates Peter Rowley and Paul Taylor
School of Mathematics The University of Manchester Alan Turing Building Oxford Road Manchester M13 6PL UK
Correspondence should be addressed to Peter Rowley peterjrowleymanchesteracuk
Received 31 July 2014 Accepted 26 November 2014
Academic Editor Peter Fleischmann
Copyright copy 2015 Chris Bates et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
For each of fifteen of the sporadic finite simple groups we determine the suborbits of its automorphism group in its conjugationaction upon its involutions Representatives are obtained as words in standard generators
1 IntroductionGroups permeate many areas of mathematics Sometimesthey have cameo roles other times they are centre stageFrequently it is involutions elements of order two that arein the spotlight For instance in the topological arena wehave involutory maps on the 3-sphere in connection with theSmith Conjecture (see eg [1ndash3]) while in Banach spaceswe encounter such things as involutive gradings and fixedpoints of involutions (see [4 5]) In areas of algebra suchas associative rings and algebraic groups there are manysightings of involutions (see [6ndash8]) Involutions can oftenhave a considerable influence on the structure of the groupto which they belong Even their absence can be tellingmdashwitness the Feit Thompson theorem [9] For a finite groupof even order Brauer and Fowler [10] establish many resultsconcerning involutions and other properties of the groupFor example they bound the index of a proper normalsubgroup in terms of the number of involutions the grouppossesses In a similar vein for a finite group with at leasttwo conjugacy classes of involutions the Thompson orderformula ([11] Theorem 351) gives its order using data closelyassociated with the involutions In the case when we havea finite nonabelian simple group more often than not itsinvolutions play a dominant role (see eg [12])
This paper studies the involutions in Aut(119870) where119870 is asmall sporadic finite simple group By small we mean that 119870is isomorphic to one of the following groups
1198721111987212 119869111987222 119869211987223119867119878 119869
311987224119872119888119871119867119890119877119906
119878119906119911 1198621199003 1198621199002
The diminutive appellation aligns more or less with saidgroup having a nontrivial permutation representation ofdegree at most 6156 Several of the larger sporadic groupshave been studied individually in [13ndash16] So for the remain-der of this paper 119870 is assumed to be a small sporadic simplegroup and119866 is a subgroup of Aut(119870) containing119870 Also 119905willdenote an involution of 119866 Put119883 = 119905
119866 the 119866-conjugacy classof 119905 Our aim is to study the suborbits of 119866 in its conjugationaction on 119883 or in other words to determine the action of119862119866(119905) on 119883 119905 This we do employing the services of the
computational algebra packages Gap [17] and Magma [18]partnered by the electronic Atlas [19] It goes without sayingtherefore that we use the Atlas notation and conventions asgiven in [20]
2 Calculating Orbit Representatives
As our starting point we take the smallest nontrivial per-mutation representation of 119866 as described in [19] with 119870
being generated by standard generators denoted here as in[19] by 119886 and 119887 In the case when 119866 = 119870 the standardgenerators for 119866 are again as in [19] denoted by 119888 and119889 Having chosen a suitable element in 119883 to play the roleof 119905 we then forage for elements 119892
1 1198922 119892
119896isin 119866 so
that 119905 1199051198921 119905119892119896 is a complete set of representatives forthe 119862
119866(119905)-orbits of 119883 In doing this we make frequent
use of the standard command IsConjugate(Hxy)mdashthisworks effectively here since the degree of the permutationrepresentation of 119866 is no more than 6156 Also observe that
Hindawi Publishing CorporationAlgebraVolume 2015 Article ID 587629 15 pageshttpdxdoiorg1011552015587629
2 Algebra
Fltabgt=FreeGroup(2)
phi=homlt F-gtG | ltaG1gtltbG2gt gt
rh=IsConjugate(Gtx)
sh=hphi
lengthmin=sh
hmin=h
Ct=Centralizer(Gt)
for c in Ct do
sch=(clowasth)phi
if sch lt lengthmin then
lengthmin=(sch)
hmin=clowasth shmin=sch
end if
end for
Algorithm 1
wemay write our conjugating element 119892119894as a word in 119886 and 119887
Thus our aim is to find 119892119894which have relatively ldquosmallrdquo length
relative to the generating set 119886 119887 for119870 However wemay notalways achieve the minimum possible length In more detailwe proceed as follows define119863
0(119886 119887) = 1 and for 119895 isin N
119863119895(119886 119887) = 119909119886 119909119887 | 119909 isin 119863
(119895minus1)(119886 119887) (1)
For 119899 isin N set
119863[119899](119886 119887) = ⋃
119895le119899
119863119895(119886 119887) (2)
Themain purpose of119863[119899](119886 119887) is to produce a colony of short
words in 119886 and 119887 a number of which may well yield thesame element of 119866 To speed matters up we prune out theseduplicates We hunt through119863
[119899](119886 119887) for 119899 typically at most
20 so as to ensnare suitable conjugating elements 119892119894 If this
does not yield enough representatives for the 119862119866(119905)-orbits
then we recalculate 119863[119899](119886 119887) with 119886 and 119887 replaced by short
words in 119886 and 119887which are also generators for119866 For examplewe might try replacing 119886 by 119886119887 and keeping 119887 the same ortry a more complicated substitution such as replacing 119887 with(119886119887)5119886 Such a substitution was used to produce a 119892
119894in the
case when 119866 cong 11987224
and 119905119909 in the class 3119860 starting fromthe word 11988711988611988721198861198871198861198872119886119887 Some of the words for the 119892
119894may be
further simplified and this was done by handOn a number of occasions this approach fails to deliver
conjugating elements for some (often of small size) 119862119866(119905)-
orbits To deal with such elusive119862119866(119905)-orbits sayO we begin
by finding an 119909 in O usually by a random search Then weobtain an element ℎ in 119866 such that 119905ℎ = 119909 Now we runthrough some or all of the elements 119888 in 119862
119866(119905) seeking a 119888 for
which 119888ℎ has small ldquolengthrdquo (ie the total number of symbols119886 119886minus1 119887 119887minus1 in the given expression for 119888ℎ) Sometimes we
may also vary ℎ In somewhat more detail the Magma codeemployed is as shown in Algorithm 1
We remark that this procedure is not usually as efficientas using the set 119863
[119899](119886 119887) and moreover usually yields more
complicated expressions for the conjugating elementsWe say a few words about how we guarantee that
we have a representative of every 119862119866(119905) orbit By using
the IsConjugate(Ctxy) command where 119905 is a fixedinvolution of the 119866-conjugacy class and Ct denotes 119862
119866(119905)
we ensure that no two representatives are in the same 119862119866(119905)-
orbitWe terminate our calculations for the119866-conjugacy classof 119905 when the sum of the 119862
119866(119905)-orbit sizes equals the size of
the conjugacy classOne may envision the information presented here to be
useful in the following circumstances Suppose 119867 is a finitegroup with 119909 119910 isin 119867 Further suppose that we have identifiedthe subgroup 119871 = ⟨119909 119910⟩ as being isomorphic to 119866 viarecognizing 119909 and 119910 as standard generators (See [21] for adiscussion of standard generators)Thenwemay translate ourinformation from 119866 to 119871 so that we see how 119871 acts upon itsinvolution conjugacy classes within the group119867 Moreover if119867 is large from a computational standpoint (eg |119867| is largeor119867 is a matrix group of large degree) then having the 119892
119894as
short words may be beneficial To facilitate applications suchas these computer files containing the 119892
119894as words in 119886 and 119887
are available on request from the second authorThe following section gives the permutation ranks of 119866
on 119883 and the elements 119892119894together with some additional
information
3 Orbit Representatives
So 119905 is a suitably chosen (and then fixed) involution of theconjugacy class119883 of 119866 For 119862 a 119866-conjugacy class we define
119883119862= 119909 isin 119883 | 119905119909 isin 119862 (3)
Since 119883119862is 119862119866(119905)-invariant 119883
119862will be a union of 119862
119866(119905)-
orbits for each 119866-conjugacy class 119862 for which 119883119862
= 0 As iswell known |119883
119862|may be calculated for any119866-conjugacy class
using the complex character table (see eg [22]) and this iseasily carried out in Gap As we proceed by breaking each119883
119862
(for |119883119862| = 0) into 119862
119866(119905)-orbits this is useful information
Our first table gives an overview of the permutation ranksof119866 in its action on119883The succeeding tables consider in turnthe possibilities for 119870 with the first column identifying 119883
119862
and the second the size of the119862119866(119905)-orbitO (contained in119883
119862)
and the third supplies a group element 119892 for which 119905119892 isin OWe emphasize that the following tables give the 119862
119870(119905)-
orbits of 119883mdashin the case when 119870 lt 119866 = Aut(119870) the tablesare annotated so as to also yield the 119862
119866(119905)-orbits of119883 Before
explaining how this is done we remark that in all instanceshere when 119870 lt 119866 = Aut(119870) we have 119862
119870(119905) lt 119862
119866(119905) Thus
119883 is a 119866-conjugacy class and 119866 = 119870119862119866(119905) Now suppose that
1198621and 119862
2are two 119870-conjugacy classes which fuse in 119866 (so
1198621cup 1198622is a 119866-conjugacy class) Then 119862
119892
1= 1198622for some
119892 isin 119862119866(119905) and consequently 119883119892
1198621
= 1198831198622
Hence if O is a119862119870(119905)-orbit contained in 119883
1198621
O cup O119892 will be a 119862119866(119905)-orbit
of 119883 In this circumstance a broken horizontal line indicatesthat 119862
1and 119862
2are fused in 119866 = Aut(119870) and if O is obtained
by using the 119894th listed conjugating element in 1198831198621
then O119892
is obtained using the 119894th listed conjugating element in 1198831198622
When a 119870-conjugacy class 119862 is also a 119866-conjugacy class itmay be the case that a 119862
119866(119905)-orbit contained in 119883
119862is the
union of two 119862119870(119905)-orbits in 119883
119862 A vertical line connecting
Algebra 3
Table 1
119866 2119860 2119861 2119862 2119863
11987211
811987212
10 11 131198691
2211987222
13 5 101198692
6 31 1411987223
9119867119878 9 27 5 321198693
27 1711987224
14 26119872119888119871 6 28119867119890 12 34 41119877119906 22 42119878119906119911 9 66 15 411198621199003
7 611198621199002
5 13 178
Table 2
119883119862
|O| 119892
2119860 12 1198871198861198873119886119887
311986024 119887
2
8 1198873119886119887
4119860 24 119887119886119887
5119860 48 119887119886119887119886119887
611986024 119887
24 1198873
Table 3
119883119862
|O| 119892
211986010
10
1003816100381610038161003816100381610038161003816100381610038161003816
11988721198861198871198861198872119886119887119886119887
119887211988611988721198861198871198861198872119886119887
2119861 15 119886
3119861 60 119886119887
4119860 30 1198861198872119886119887119886
4119861 30 1198871198861198871198861198871198861198872119886
5119860 120 119887119886
611986060
60
1003816100381610038161003816100381610038161003816100381610038161003816
119887
1198872
two 119862119870(119905)-orbits in 119883
119862signifies that their union is a 119862
119866(119905)-
orbit
31 Permutation Ranks See Table 1
32 119870 cong 11987211
321119883 = 2119860 119905 = 119886 See Table 2
33 119870 cong 11987212
331 119883 = 2119860 119905 = ((119886119887)21198861198872)3 See Table 3
Table 4
119883119862
|O| 119892
211986124 119887119886119887
2119886119887
6 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886119887119886119887
311986032
32
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887
1198871198861198872
3119861 16 11988711988611988721198861198872119886119887119886119887
4119860 48 1198871198861198871198861198872
4119861 48 1198872119886119887119886119887
5119860 96 119887119886119887
611986196
96
1003816100381610038161003816100381610038161003816100381610038161003816
119887
1198872
Table 5
119883119862
|O| 119892
211986015 119887119886119887
2119886119887119886119887119886119887
1 119887119886119887119886119887119886119887211988611988721198861198871198861198872119886119887
2119861 15 1198871198861198871198861198872
311986020
20
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887
1198861198871198861198872119886119887119886
3119861 60 1198871198861198872
511986060 119886119887119886119887
2119886
60 119886119887
6119860 60 119887119886119887119886
611986160
60
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887
119886
1011986060 119887119886119887
2119886119887119886
60 119887119886119887119886119887
11119860 120 119887
11119861 120 119887119886
332 119883 = 2119861 119905 = 119886 See Table 4
333 119883 = 2119862 119905 = 119888 See Table 5
34 119870 cong 1198691
341 119883 = 2119860 119905 = 119886 See Table 6
35 119870 cong 11987222
351 119883 = 2119860 119905 = 119886 See Table 7
352 119883 = 2119861 119905 = 119888 See Table 8
353 119883 = 2119862 119905 = (1198881198891198881198892)5 See Table 9
36 119870 cong 1198692
361 119883 = 2119860 119905 = (1198871198861198872(119886119887)2119887(119886119887)3)5 See Table 10
4 Algebra
Table 6
119883119862
|O| 119892
211986015 119887119886119887
2119886119887119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
15 119887119886119887211988611988721198861198871198861198872119886119887119886119887119886119887
2119886119887119886119887
311986060 119887119886119887
2119886119887119886119887119886119887
2119886119887
12 1198872119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
511986060 119887119886119887119886119887
21198861198872119886119887119886119887
20 1198872119886119887119886119887119886119887
21198861198872119886119887
511986160 119887119886119887119886119887
21198861198871198861198872119886119887119886119887
20 11988721198861198871198861198872119886119887119886119887
611986060 119887119886119887
2119886119887119886119887
60 1198871198861198871198861198872119886119887
7119860 120 11988711988611988721198861198872119886119887119886119887
1011986060 119887
2119886119887119886119887119886119887
60 1198871198861198871198861198871198861198872
1011986160 119887
2119886119887119886119887
60 1198871198861198871198861198872
11119860 120 1198871198861198871198861198872119886119887119886119887
15119860 120 11988721198861198872119886119887119886119887
15119861 120 119887119886119887
19119860 120 119887
19119861 120 1198872119886119887
19119862 120 11988721198861198871198861198872119886119887
Table 7
119883119862
|O| 119892
2119860
24 1198871198861198873119886119887
12 (119887119886)21198872119886119887119886119887
8 11988721198861198871198861198872(119886119887)3
6 1198873(119886119887)2(1198861198873)2119886119887
3119860 192 119887119886119887
4119860
48 1198871198861198872119886119887
48 1198873119886119887119886119887
48 1198871198861198871198861198873
411986196 119887
2119886119887119886119887
96 1198871198861198871198861198872
5119860 384 1198872
6119860 192 119887
Table 8
119883119862
|O| 119892
211986042 119887
7 (1198872119886)2119887119886
3119860 112 119886
4119860 168 119886119887119886119887119886
Table 9
119883119862
|O| 119892
211986020 119887
2
5 11988711988611988721198861198873
3119860 80 119887
4119860 80 119887119886119887119886119887119886
411986140 119887119886119887
2119886
40 1198871198861198872119886119887
5119860 320 119886119887
611986080
80
1003816100381610038161003816100381610038161003816100381610038161003816
119886
1198861198872119886
11119860 320 1198872119886
11119861 320 119887119886
Table 10
119883119862
|O| 119892
2119860 10 119887119886
3119861 160 119887
4119860 80 119886
5119860 32 1198861198872119886119887119886
5119861 32 1198871198861198872
362 119883 = 2119861 119905 = 119886 See Table 11
363 119883 = 2119862 119905 = 119888 See Table 12
37 119870 cong 11987223
371119883 = 2119860 119905 = 119886 See Table 13
38 119870 cong 119867119878
381 119883 = 2119860 119905 = 119886 See Table 14
382 119883 = 2119861 119905 = ((119886119887)2119886)5 See Tables 15 and 16
383 119883 = 2119862 119905 = 119888 See Table 17
384 119883 = 2119863 119905 = ((119889119888)21198894119888119889)5 See Tables 18 and 19
39 119870 cong 1198693
391119883 = 2119860 119905 = 119886 See Table 20
392119883 = 2119861 119905 = 119888 See Table 21
310 119870 cong11987224
3101 119883 = 2119860 119905 = ((119886119887)31198861198872)4 See Tables 22 and 23
Algebra 5
Table 11
119883119862
|O| 119892
2119860 15 119887(119887119886)2119887(119887119886)
2119887(119887119886)
2
2119861
15
15
1003816100381610038161003816100381610038161003816100381610038161003816
((119887119886)2119887)3(119887119886)3
119887(119887119886)2(119887(119887119886)
31198872119886)2
1
1
1003816100381610038161003816100381610038161003816100381610038161003816
(119887119886)21198872119886(119887(119887119886)
2)21198872119886119887(119887119886)
3
1198872119886119887(119887119886)
3119887(119887119886)
2119887(119887119886)
3(1198872119886)2
3119860 24 119887(119887119886)2119887(119887119886)
2
3119861 60 119887(119887119886)3119887(119887119886)
2119887(119887119886)
3119887(119887119886)
4119860 60 (119887119886)119887(119887119886)2
5119860 60 (119887119886)2119887(119887119886)
2119887(119887119886)
3119887(119887119886)119887(119887119886)
2
5119861 60 (119887119886)119887(119887119886)119887(119887119886)3119887(119887119886)
2119887(119887119886)
3
5119862120 119887(119887119886)
3119887(119887119886)
3
24 119887(119887119886)119887(119887119886)2119887(119887119886)
3
5119863120 (119887119886)119887(119887119886)
2119887(119887119886)
3
24 (119887119886)2119887(119887119886)
2119887(119887119886)119887(119887119886)
6119860 120 (119887119886)3
611986160
60
1003816100381610038161003816100381610038161003816100381610038161003816
(119887119886)2119887(119887119886)
3
(119887119886)3119887(119887119886)
4
7119860 240 119887(119887119886)4
1011986060 119887(119887119886)119887(119887119886)
2119887(119887119886)
2
60 119887(119887119886)2119887(119887119886)
3
1011986160 (119887119886)
2119887(119887119886)
2119887(119887119886)
60 (119887119886)119887(119887119886)2119887(119887119886)119887(119887119886)
10119862120 (119887119886)119887(119887119886)119887(119887119886)
120 (119887119886)2119887(119887119886)
10119863120 119887(119887119886)
3
120 119887(119887119886)119887(119887119886)2
12119860120
120
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119887119886)
(119887119886)
15119860 240 (119887119886)119887(119887119886)119887(119887119886)2119887(119887119886)
2
15119861 240 (119887119886)119887(119887119886)2119887(119887119886)119887(119887119886)
2
Table 12
119883119862
|O| 119892
2119860 21 (1198871198861198872119886)2(119887119886)3
2119861 28 1198871198861198872
311986014
14
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)3(1198861198872)2119886119887
(1198872119886119887119886)2119887119886119887
3119861 168 119887119886119887119886
411986084 119886119887119886119887
2119886119887
42 1198872119886119887119886119887
6119860
84
84
1003816100381610038161003816100381610038161003816100381610038161003816
1198872
119887
42
42
1003816100381610038161003816100381610038161003816100381610038161003816
119886119887119886119887119886119887119886
1198861198871198861198872119886119887119886119887
6119861 168 119886119887
7119860 336 119887119886119887
8119860168 119886119887
2119886
168 119886119887119886
12119860168
168
1003816100381610038161003816100381610038161003816100381610038161003816
119887119886
119886
Table 13
119883119862
|O| 119892
211986084 119887119886119887
3119886119887
14 11988731198861198872119886119887
3119860 448 1198872
4119860
336 119887119886119887119886119887
336 119887119886119887119886119887119886119887
336 1198871198861198871198861198871198861198872
5119860 896 119887119886119887
6119860 1344 119887
Table 14
119883119862
|O| 119892
211986080 119887
211988611988731198861198872
30 119887211988611988711988611988721198861198871198861198872
3119860 640 119887119886119887
4119861 480 1198871198861198873119886119887
4119862 960 1198872119886119887
5119861 128 1198871198861198872119886119887
5119862 1536 119887
6119861 1920 1198872
Table 15
119883119862
|O| 119892
211986045 119887119886119887
3119886119887
30 11988741198861198874119886119887119886119887
211986136
36
1003816100381610038161003816100381610038161003816100381610038161003816
119887211988611988731198861198872
1198871198861198871198861198871198861198873119886
3119860360 119887
31198861198872119886119887119886119887
24 1198861198872119886119887211988611988711988611988721198861198872119886
4119861 180 119886
4119862
360 11988711988611988711988611988741198861198872119886
180
180
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198871198861198872
1198872119886119887119886119887119886119887
5119860144
144
1003816100381610038161003816100381610038161003816100381610038161003816
11988711988611988741198861198873119886119887
11988711988611988731198861198874119886119887
5119861 720 1198871198861198871198861198872119886
Table 16
119883119862
|O| 119892
6119861
720 119887119886119887
360 1198871198861198872119886
360 1198861198872119886119887
7119860 2880 119887
8119861720 119887
41198861198872119886
720 1198861198873119886119887
8119862720 119887119886119887
3119886
720 11988611988721198861198874
10119860720
720
1003816100381610038161003816100381610038161003816100381610038161003816
1198872
1198873
10119861720
720
1003816100381610038161003816100381610038161003816100381610038161003816
119886119887119886
119887119886
15119860 2880 1198872119886
6 Algebra
Table 17
119883119862
|O| 119892
2119860 105 119886
2119861 28 119887119886
3119860 336 1198872
4119861 630 119887
Table 18
119883119862
|O| 119892
211986060 119886119887
31198861198871198861198872119886
5 11988611988731198861198872119886
2119861 60 1198872119886
3119860240 119886
160 11988611988731198861198871198861198872119886119887119886
4119860 120 1198871198861198873119886119887119886
4119861
240 119886119887
120 1198871198861198872119886119887119886119887
30 1198861198872119886
4119862120 119886119887
21198861198872119886
120 1198861198873119886119887119886
5119860192 119886119887119886119887
3
192 11988711988611988721198861198873
5119861 320 1198874119886119887
5119862 1920 11988721198861198872
6119860480 119886119887
2119886119887
480 1198871198861198872119886
Table 19
119883119862
|O| 119892
6119861480 119887
4119886119887119886119887
480 11988721198861198872119886
7119860 1920 119887
8119860480 119887119886119887119886119887
2
480 11988711988611988721198861198872119886119887
10119860960 119887
211988611988721198861198872119886
960 119887119886119887
10119861 960 1198861198872
11119860 1920 1198872119886119887119886
11119861 1920 119887119886119887119886119887
12119860960 119887119886
960 1198873
15119860 1920 1198871198861198873
20119860960 119887119886119887119886
960 1198872119886119887
20119861960 119887
21198861198873119886
960 1198874119886119887119886
Table 20
119883119862
|O| 119892
2119860120 119887(119887119886)
3119887(119887119886)119887(119887119886)
5
10 (119887119886)119887(119887119886)4119887(119887119886)
2
3119860 192 119887(119887119886)6
3119861 640 (119887119886)119887(119887119886)2
4119860480 119887(119887119886)
5119887(119887119886)
80 (119887119886)3119887(119887119886)
3119887(119887119886)
4
5119860960 (119887119886)
3119887(119887119886)
3119887(119887119886)
2
320 (119887119886)2119887(119887119886)
2
5119861960 (119887119886)119887(119887119886)
3119887(119887119886)
4
320 (119887119886)119887(119887119886)3
6119860 960 (119887119886)119887(119887119886)6
8119860960 (119887119886)
4119887(119887119886)
2
960 (119887119886)119887(119887119886)5
9119860 1920 119887(119887119886)2119887(119887119886)
2
9119861 1920 (119887119886)
9119862 1920 119887(119887119886)2
10119860960 (119887119886)
4119887(119887119886)
960 119887(119887119886)4119887(119887119886)119887(119887119886)
10119861960 119887(119887119886)119887(119887119886)119887(119887119886)119887(119887119886)
2
960 119887(119887119886)5
12119860960
960
1003816100381610038161003816100381610038161003816100381610038161003816
(119887119886)2119887(119887119886)
4
(119887119886)6
15119860 1920 (119887119886)2119887(119887119886)
15119861 1920 119887(119887119886)3
17119860 1920 (119887119886)2
17119861 1920 (119887119886)2119887(119887119886)
3
Table 21
119883119862
|O| 119892
2119860 153 1198861198872119886119887119886119887119886119887
3119860102
102
1003816100381610038161003816100381610038161003816100381610038161003816
1198872
119887
3119861 272 1198861198871198861198872119886
4119860612 119887119886119887119886119887119886119887
2119886
306 1198871198861198872119886119887119886119887119886
6119860
612
612
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198872119886119887119886
1198871198861198871198861198872119886
306
306
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198872119886119887119886119887119886
119887119886119887119886
81198601224 119886119887119886119887119886119887
1224 1198872119886119887119886119887119886
9119860 816 11988721198861198871198861198872119886119887
9119861 816 1198871198861198871198861198872119886119887
9119862 816 1198871198861198872
121198601224
1224
1003816100381610038161003816100381610038161003816100381610038161003816
119886119887
1198861198872119886119887119886
17119860 2448 119887119886
17119861 2448 119886
19119860 2448 1198861198872119886119887
19119861 2448 1198872119886119887119886
Algebra 7
Table 22
119883119862
|O| 119892
2119860
168 1198872119886119887119886119887119886119887
2119886119887
14 119886119887119886
14 1198861198871198861198872119886119887
2119861 84 1198872119886119887
3119860 896 119887
3119861 128 119887211988611988721198861198871198861198872119886119887119886119887
Table 23
119883119862
|O| 119892
4119860112 119887119886119887119886119887119886119887
112 1198872119886119887211988611988721198861198872
4119861
1344 1198861198872119886119887119886
672 11988611988721198861198871198861198872119886119887
672 11988721198861198871198861198872119886119887119886
5119860 1792 119887119886119887119886
6119860 5376 119886
Table 24
119883119862
|O| 119892
211986060 119887
2119886119887119886119887211988611988721198861198871198861198872119886119887119886119887119886119887
2119886119887
15 11988711988611988721198861198872119886119887119886119887119886119887
2
2119861
120 119887119886119887211988611988711988611988721198861198871198861198872
80 119887211988611988711988611988721198861198871198861198872119886119887
2 (119887119886)8(1198872119886)21198871198861198872(119886119887)2(1198861198872)2(119886119887)3(1198861198872119886119887)2119887
3119860 320 119887211988611988721198861198871198861198872119886119887119886119887119886119887
3119861 960 11988721198861198871198861198872119886119887
4119860120 119887119886119887
21198861198872
120 1198871198861198871198861198872
4119861
480 1198872119886119887119886119887119886119887
2119886119887
240 1198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
240 1198872119886119887119886119887119886119887119886119887119886119887119886119887
4119862
960 1198871198861198872119886119887
960 11988721198861198871198861198872
160 1198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
21198861198872
160 1198872119886119887119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
5119860 1920 119887119886119887
6119860 1920 1198871198861198871198861198871198861198872
61198611920 119887119886119887
2
1920 1198872119886119887
101198601920 119887
21198861198871198861198872119886119887119886119887
1920 1198872119886119887119886119887119886119887119886119887
2
11119860 7680 119887119886119887119886119887
121198613840 119887
3840 1198872
3102 119883 = 2119861 119905 = 119886 See Table 24
311 119870 cong 119872119888119871
3111119883 = 2119860 119905 = 119886 See Table 25
3112119883 = 2119861 119905 = 119888 See Table 26
Table 25
119883119862
|O| 119892
2119860 210 119887211988611988731198861198872
3119861 2240 1198871198861198871198861198872
4119860 5040 1198872119886119887
5119861 8064 119887
6119861 6720 1198872
Table 26
119883119862
|O| 119892
2119860 165 11988611988721198861198871198861198872119886119887
3119860 220 1198872119886119887311988611988721198861198871198861198872119886
3119861660 119886119887
21198861198872
220 1198872119886119887119886119887119886119887
3
41198601980 119887
21198861198873119886119887119886119887119886
990 119886
5119860792 119887
211988611988721198861198873
792 1198872119886119887119886119887119886119887
2
5119861 7920 1198872
6119860 1980 119886119887
6119861
1980 1198871198861198872119886119887119886119887
1980 1198861198872119886119887
660
660
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198873
11988711988611988731198861198874119886119887
71198603960 119887119886119887119886119887
2119886
3960 1198871198861198872
71198613960 119887119886119887
3119886
3960 1198872119886
81198603960 119887
21198861198872119886
3960 11988611988731198861198873
9119860 2640 1198872119886119887119886
9119861 2640 1198861198871198861198872119886119887
101198603960 119887119886119887119886119887
3960 119886119887119886119887119886
121198603960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198873
1198871198861198873
141198603960 119886119887
21198861198873
3960 1198871198861198872119886119887
141198613960 119887119886119887119886119887119886
3960 1198872119886119887119886119887119886
151198603960 119887119886119887
3960 1198874
151198613960 119886119887119886119887
3960 119887
301198603960 119887119886
3960 1198871198861198872119886
301198613960 119887
41198861198872119886
3960 1198861198872
8 Algebra
Table 27
119883119862
|O| 119892
2119860240 119887119886119887119886119887119886119887
2 1198875(1198861198876)2(119886119887)31198875119886(11988721198861198873)2
2119861 315 119887211988611988711988611988721198861198872119886119887
3119860 1344 1198872119886119887
3119861 960 1198871198861198874119886119887
4119860480 119887
31198861198871198861198872119886119887
480 11988721198861198871198861198873119886119887
41198612520
2520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887
1198874119886119887
5119860 2688 1198873
6119860 13440 119887
Table 28
119883119862
|O| 119892
2119860 84 (119886119887)21198875(119886119887)2
2119861
168
168
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)2
(119886119887)1198874(119886119887)119887
2(119886119887)119887
2(119886119887)119887
4
14
14
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198874(119886119887)119887
4(119886119887)119887
2(119886119887)119887
2
1198872(119886119887)119887
2(119886119887)21198874(119886119887)119887
2(119886119887)119887
2
3119860 896 1198874(119886119887)119887
2
3119861 2688 1198872
4119861
1344
1344
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)119887
2(119886119887)119887
2
1198872(119886119887)119887
4(119886119887)21198874
112
112
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)1198872(119886119887)41198874(119886119887)2
(119886119887)31198874(119886119887)41198872
4119862
1344
1344
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)1198874(119886119887)119887
4
1198874(119886119887)119887
2(119886119887)119887
2(119886119887)31198872
672
672
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198872(119886119887)3
1198872(119886119887)41198872(119886119887)2
672
672
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)41198872(119886119887)2
(119886119887)4119887(119886119887)
2
5119860 5376 (119886119887)3
6119860 5376 (119886119887)1198872(119886119887)119887
2(119886119887)4
61198615376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198872(119886119887)
(119886119887)1198872(119886119887)
7119862 3072 1198872(119886119887)119887
2(119886119887)119887
2(119886119887)2
81198605376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)21198872(119886119887)21198872
(119886119887)1198872(119886119887)21198872(119886119887)
101198605376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)2
1198872(119886119887)119887
2(119886119887)119887
2
1211986110752
10752
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)2
1198874(119886119887)2
15119860 21504 1198872(119886119887)3
17119860 21504 (119886119887)1198872(119886119887)2
17119861 21504 (119886119887)
21119860 21504 (119886119887)1198872(119886119887)119887
2
21119861 21504 1198872(119886119887)
Table 29
119883119862
|O| 119892
2119860105 (119886119887)119887(119886119887)
3119887(119886119887)
3
63 119887611988611988731198861198873119886119887211988611988761198861198872119886119887611988611988761198861198874119886
2119861 315 (1198861198872)2(119887119886)21198872119886
3119860630 (119886119887)119887(119886119887)119887(119886119887)119887
126 119887(119886119887)21198876
3119861 2520 (119886119887)1198876
41198601260 (119886119887)
21198875(119886119887)
1260 (119886119887)119887(119886119887)1198873
4119861
945
945
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)21198872(119886119887)119887
119887(119886119887)119887(119886119887)1198872(119886119887)119887
315
315
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119886119887)5119887
119887(119886119887)31198872(119886119887)119887
41198621890
1890
1003816100381610038161003816100381610038161003816100381610038161003816
1198873(119886119887)2
(119886119887)1198872(119886119887)2119887
5119860 2520 (119886119887)3119887(119886119887)
6119860
3780 (119886119887)31198872(119886119887)2
1890 119887(119886119887)119887(119886119887)
1890 119887(119886119887)119887(119886119887)119887(119886119887)3
1260 119887(119886119887)1198872(119886119887)119887(119886119887)
2
6119861 7560 (119886119887)
71198601890 (119886119887)119887(119886119887)
41198872
90 119887(119886119887)1198873(119886119887)2119887
71198611890 119887
5(119886119887)119887(119886119887)
90 1198875(119886119887)119887
5(119886119887)119887
2(119886119887)119887
5
7119862 2160 119887(119886119887)119887(119886119887)4
71198637560 (119886119887)119887(119886119887)
4
1080 (119886119887)1198873(119886119887)
71198647560 (119886119887)
3119887(119886119887)
2
1080 (119886119887)1198874(119886119887)2119887(119886119887)
8119860
3780
3780
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)2119887(119886119887)
2
(119886119887)4
3780
3780
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)3119887(119886119887)119887(119886119887)119887
(119886119887)119887(119886119887)119887(119886119887)2
10119860 7560 (119886119887)3
121198607560 119887(119886119887)119887
7560 119887(119886119887)1198872
121198617560
7560
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119886119887)2119887
119887
14119860
3780 (119886119887)1198873(119886119887)2
3780 (119886119887)21198872(119886119887)
1890 (119886119887)1198872(119886119887)
1890 119887(119886119887)3119887
14119861
3780 (119886119887)119887(119886119887)2119887(119886119887)
3780 (119886119887)1198874
1890 (119886119887)21198872(119886119887)119887(119886119887)
2
1890 119887(119886119887)119887(119886119887)2
Algebra 9
Table 29 Continued
119883119862
|O| 119892
141198627560 119887(119886119887)
7560 119887(119886119887)3
141198637560 119887
3(119886119887)
7560 119887(119886119887)2
15119860 15120 (119886119887)1198872
21119860 15120 (119886119887)4119887(119886119887)
21119861 15120 (119886119887)119887
21119862 15120 (119886119887)3119887
21119863 15120 1198872
281198607560 119887
2(119886119887)119887
2
7560 (119886119887)2
281198617560 119887
4(119886119887)3
7560 (119886119887)119887(119886119887)2
Table 30
119883119862|O| 119892
2119860
640 11988611988731198861198872119886119887119886
480 11988611988721198861198872119886119887119886119887119886119887119886119887119886119887
21198861198872119886
240 11988611988721198861198873119886119887311988611988731198861198873119886119887211988611988711988611988721198861198873119886119887119886119887119886119887119886119887
31198861198873
30 1198861198872119886119887119886119887119886119887211988611988721198861198873119886119887311988611988711988611988711988611988721198861198873119886119887211988611988721198861198873119886119887211988611988711988611988711988611988731198863119860 10240 119886119887
3119886119887119886119887119886119887119886119887119886119887
2119886
4119860 5120 119886119887119886119887119886119887119886
4119862
7680 119886119887119886119887211988611988721198861198872119886119887119886
3840 119886119887211988611988731198861198871198861198872119886119887119886
3840 119886119887119886119887211988611988711988611988731198861198872119886
411986315360 119886119887119886
7680 1198861198872119886119887211988611988711988611988721198861198871198861198872119886
5119860 12288 11988611988731198861198871198861198872119886119887119886119887119886119887119886
5119861 4096 119886119887119886119887119886
611986061440 119886119887
21198861198871198861198872119886119887119886
30720 119886119887119886119887119886119887119886119887119886119887119886
7119860 61440 119886
811986261440 119886119887119886119887
21198861198872119886
61440 11988611988721198861198872119886119887119886
10119860 61440 11988611988721198861198872119886119887119886119887119886119887119886
12119860122880 1198861198872119886119887119886
13119860 61440 1198861198873119886119887119886
312 119870 cong 119867119890
3121119883 = 2119860 119905 = 119886 See Table 27
3122119883 = 2119861 119905 = (1198861198872)6 See Table 28
3123 119883 = 2119862 119905 = 1198893 See Table 29
313 119870 cong 119877119906
3131 119883 = 2119860 119905 = 1198872 See Table 30
3132 119883 = 2119861 119905 = 119886 See Table 31
314 119870 cong 119878119906119911
3141119883 = 2119860 119905 = ((119886119887)21198861198872)6 See Table 32
Table 31
119883119862 |O| 119892
2119860 455 119887119886119887311988611988721198861198873119886119887
2119861
455 11988711988611988731198861198872119886119887119886119887119886119887119886119887119886119887
455 11988711988611988711988611988711988611988711988611988711988611988721198861198873119886119887
1 1198873119886119887311988611988731198861198872119886119887119886119887119886119887
211988611988721198861198873119886119887119886119887211988611988721198861198872
1 1198871198861198872119886119887119886119887311988611988711988611988721198861198872119886119887119886119887211988611988711988611988731198861198873119886119887119886119887119886119887
31198861198871198861198872119886119887119886119887211988611988731198861198871198861198873
3119860 5824 1198872119886119887119886119887119886119887119886119887119886119887
4119860 3640 (119886119887)10119886(119886119887)15119886((119886119887)
5119886)2
4119862 3640 1198873119886119887119886119887119886119887
21198861198872
4119863 3640 119887311988611988711988611988721198861198873119886119887
51198605824 119887
311988611988711988611988721198861198871198861198872
5824 119887211988611988731198861198872119886119887
5119861 29120 1198872119886119887119886119887
6119860 29120 1198872119886119887119886119887119886119887
2119886119887
7119860
29120 11988721198861198872
29120 119887
29120 1198872
1011986029120 119887119886119887119886119887
2119886119887119886119887
29120 119887119886119887119886119887
1011986129120 119887119886119887119886119887
31198861198873119886119887
29120 11988711988611988731198861198873119886119887119886119887
12119860 58240 1198873119886119887119886119887119886119887
2119886119887
13119860
29120 119887119886119887
29120 119887119886119887119886119887119886119887
29120 11988731198861198872119886119887
1411986029120 119887
311988611988721198861198872119886119887119886119887
29120 11988711988611988721198861198872119886119887119886119887119886119887
1411986129120 119887
21198861198872119886119887119886119887119886119887
29120 11988711988611988711988611988711988611988721198861198872
1411986229120 119887119886119887
2119886119887119886119887119886119887119886119887
2
29120 1198872119886119887119886119887119886119887119886119887
2119886119887
15119860 116480 11988731198861198872119886119887119886119887119886119887
2011986058240 119887
31198861198871198861198872
58240 11988721198861198873119886119887
2611986029120 119887119886119887
2
29120 1198872119886119887
2611986129120 119887119886119887
21198861198872
29120 11988721198861198872119886119887
2611986229120 119887
3119886119887119886119887119886119887
29120 11988711988611988711988611988721198861198872
29119860 116480 1198871198861198872119886119887
29119861 116480 1198873119886119887119886119887
Table 32
119883119862
|O| 119892
2119860360 119886
54 ((119886119887)2119886)4119886((119886119887)
2119886)3
3119861 5120 11988721198861198872119886119887119886119887119886119887119886119887119886119887119886
3119862 9216 119887119886119887119886
4119860 1728 119887211988611988721198861198872119886119887119886119887119886119887
4119862 17280 1198872119886119887
5119861 55296 1198872119886119887119886119887119886
6119863 46080 119887
10 Algebra
Table 33119883119862|O| 119892
2119860 315 1198871198861198872119886119887119886119887119886119887
21198861198872119886119887211988611988721198861198872119886119887119886119887119886119887
2119886119887
2119861
630 1198872119886119887119886119887119886119887119886119887
2119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
21198861198872119886119887119886119887211988611988721198861198872119886119887
560 119887119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
2 (1198871198861198872119886)2(1198872119886)2(119887119886(1198872119886)2)2(1198872119886119887119886)2((1198872119886)21198872119886119887119886)21198871198861198872(119886119887)2
3119860 224 (1198871198861198872119886)3119887(119887119886)3(1198872119886)2(119887119886)5119887(119887119886)4(1198872119886)2119887
3119861 2240 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887
3119862 20160 11988711988611988721198861198871198861198872119886119887
41198602520 119887119886119887119886119887119886119887
2119886119887211988611988721198861198872119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886119887119886119887
4119861
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
2520 11988711988611988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
21198861198872119886119887
1260 119887211988611988721198861198871198861198872119886119887119886119887
4119862
5040 11988711988611988721198861198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
2520 11988711988611988711988611988711988611988721198861198872119886119887119886119887119886119887
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
4119863
10080 11988711988611988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
10080 11988711988611988711988611988721198861198872119886119887119886119887211988611988721198861198872119886119887
1120 (119886119887)4((119886119887)4119886)4119886(119886119887)8((119886119887)4119886)2
1120 (119886(1198872119886)4)2(1198872119886)8119886(119886(1198872119886)4)4(1198872119886)4
5119860 20160 119887119886119887119886119887
5119861 40320 1198872119886119887119886119887119886119887
2119886119887
6119860 10080 1198871198861198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
6119863 20160 11988711988611988721198861198872119886119887
611986440320 119887
2119886119887119886119887119886119887
2119886119887119886119887119886119887
40320 1198872119886119887119886119887211988611988721198861198872119886119887119886119887119886119887
7119860 80640 1198872119886119887119886119887119886119887119886119887
811986020160 119887119886119887
2119886119887119886119887211988611988721198861198872119886119887
20160 1198871198861198872119886119887211988611988721198861198871198861198872119886119887
8119861
20160 1198871198861198872119886119887119886119887119886119887119886119887
211988611988721198861198871198861198872119886119887
20160 11988711988611988711988611988721198861198872119886119887
20160 11988711988611988721198861198872119886119887119886119887
20160 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887119886119887
2119886119887
10080 11988721198861198872119886119887
10080 1198872119886119887119886119887
8119862
20160 11988721198861198871198861198872119886119887119886119887119886119887
20160 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
2119886119887
20160 1198872119886119887211988611988721198861198871198861198872119886119887
20160 1198871198861198872119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
10119860
40320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
119887211988611988711988611988721198861198871198861198872119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886119887
20160 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887
20160 11988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
1011986140320 119887119886119887119886119887119886119887119886119887
2119886119887
40320 1198871198861198872119886119887119886119887119886119887119886119887
11119860 161280 11988721198861198871198861198872119886119887119886119887
1211986040320 119887
2119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
20160 119887211988611988711988611988721198861198872119886119887119886119887119886119887
2119886119887
1211986140320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886119887
11988721198861198872119886119887211988611988721198861198872119886119887
12119862 40320 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
1211986380640 119887119886119887
2119886119887119886119887119886119887
80640 1198871198861198871198861198871198861198872119886119887
13119860 161280 119887119886119887
13119861 161280 11988721198861198872119886119887119886119887
14119860 80640 119887211988611988711988611988721198861198872119886119887119886119887
15119860 161280 119887
15119861 161280 1198872119886119887
15119862 161280 119887211988611988721198861198871198861198872119886119887119886119887119886119887
Table 33 Continued119883119862
|O| 119892
2011986080640
80640
1003816100381610038161003816100381610038161003816100381610038161003816
119887119886119887211988611988721198861198872119886119887119886119887
119887119886119887119886119887211988611988721198861198872119886119887
21119860 161280 119887119886119887211988611988721198861198872119886119887119886119887119886119887
21119861 161280 119887119886119887211988611988721198861198872119886119887
2411986080640 119887119886119887119886119887
2119886119887
80640 1198871198861198872119886119887119886119887
Table 34
119883119862
|O| 119892
2119860 315 ((119886119887)3119886)2119886((119886119887)
3119886)4119886(119886119887)
3((119886119887)3119886)2119886
2119861 1800 1198861198871198861198871198861198872
3119860 200 119887211988611988721198861198871198861198872119886119887119886119887
3119862 16800 11988611988721198861198872119886119887119886119887119886
41198603150 119887119886119887
21198861198871198861198872119886
630 1198871198861198872119886119887119886119887119886119887119886119887119886119887
2119886
4119862 12600 1198871198861198872119886119887119886
5119861 20160 119886119887119886
6119860 12600 1198861198871198861198871198861198872119886119887
6119864 100800 119887119886
7119860 100800 119887
811986025200 119886119887119886119887119886119887
25200 119887119886119887119886119887119886119887
12119860 50400 119886
Table 35
119883119862
|O| 119892
2119860 495 (119886119887)5119886 (119886119887)
2(119886 (119886119887)
5)4
(119886119887)5
2119861 792 11988721198861198871198861198872119886119887119886119887119886119887
2119886119887119886119887
3119860 264 119886(119886119887)9((119886119887)9119886)4119886(119886119887)
9((119886119887)9119886)2
31198615280 119886(119886119887)
6((119886119887)3119886)4
1760 (119886((119886119887)3119886)2)4119886
3119862 2640 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886
4119860
2970 11988711988611988711988611988721198861198871198861198872119886119887119886
495
495
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886
4119861 11880 119887119886119887211988611988721198861198871198861198872119886119887119886
4119862 11880 1198872119886119887
5119860 19008 119887119886119887119886
5119861 47520 1198861198872119886119887119886119887119886
6119860
11880 119886119887119886
3960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198871198861198872119886
11988611988711988611988721198861198872119886
611986115840 119886119887
2119886119887119886119887
15840 1198861198872119886119887119886119887119886119887119886119887119886119887
611986215840 119887
21198861198872119886119887119886
15840 1198872119886119887119886119887119886119887119886119887119886119887
21198861198872
611986347520 119886119887
2119886119887119886119887119886119887119886
15840 1198861198872119886119887119886119887119886119887
2119886119887119886
Algebra 11
Table 35 Continued
119883119862
|O| 119892
6119864 15840 119886119887119886119887119886119887119886119887211988611988711988611988721198861198872119886119887
7119860 47520 1198871198861198872119886119887119886
811986147520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886
119886119887119886119887119886119887
911986095040 119887119886119887119886119887119886119887
31680 119887119886119887119886119887211988611988721198861198871198861198872119886
911986195040 119887
2119886119887119886119887119886
31680 1198861198872119886119887119886119887119886119887119886119887119886119887119886
10119860 95040 119886119887
1011986147520 119887119886119887119886119887
21198861198872119886
47520 11988711988611988721198861198872119886119887119886
11119860 190080 119886
12119860 47520 1198872119886119887119886
12119861
47520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198872119886119887119886119887119886119887
2
119886119887119886119887119886119887119886119887119886
15840
15840
1003816100381610038161003816100381610038161003816100381610038161003816
11988721198861198872119886119887119886119887
119887119886119887119886119887119886
1211986247520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198872119886119887119886
1198861198871198861198871198861198871198861198871198861198871198861198872
12119864
47520 11988721198861198872119886
47520 119886119887119886119887
47520 119887119886119887119886119887119886119887119886119887
47520 1198872119886119887119886119887119886119887119886
1411986047520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198871198861198872119886119887119886119887
119887
15119862 190080 119886119887119886119887119886
1811986095040 119887119886119887
2119886119887119886119887119886
95040 11988611988721198861198872119886119887119886119887
1811986195040 119887
21198861198872119886119887119886119887119886
95040 1198871198861198871198861198871198861198872119886
2011986095040
95040
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198872119886119887119886
119887119886
Table 36
119883119862
|O| 119892
2119860 630 119886119887211988621198872119886
3119861 8960 11988611988721198862
3119862 1920 11988621198871198862
Table 37
119883119862
|O| 119892
4119861 30240 1198861198872119886
5119861 48384 1198861198871198862
6119862 80640 119886
Table 38
119883119862
|O| 119892
2119860 495 119887119886119887119886211988711988621198871198861198872119886119887
2119861396 119886119887
311988611988721198862119887119886
396 11988621198873119886119887211988621198871198862
31198601320 119887119886
211988711988621198872119886119887
440 11988611988721198861198871198861198872119886119887119886119887
31198612640 119886
211988721198861198871198861198873119886
1584 119886119887119886119887119886211988731198862
3119862 7920 11988711988611988721198861198872119886119887
4119860 2970 1198871198861198871198861198871198861198872119886119887119886
4119861
11880 1198862119887
5940 1198872
2970 1198871198861198871198861198872119886119887
51198609504 119887
21198861198871198862
9504 119887119886211988731198862119887119886
5119861 47520 11988621198872119886
611986011880 119887119886
211988721198861198872119886
3960 119886119887211988621198871198861198871198862119887119886
611986115840 119886
2
15840 119886
6119862
23760 1198871198861198871198861198873119886
23760 11988711988621198872119886119887119886
15840 1198871198861198873119886119887119886
611986423760 119887
2119886
23760 11988621198872
711986095040 119887119886
2
31680 1198871198861198872119886119887119886
811986023760 119887119886119887119886119887
2119886
23760 1198872119886119887119886119887119886119887
811986123760 119886119887119886119887
2119886
23760 11988721198862119887119886
3142 119883 = 2119861 119905 = 119886 See Table 33
3143 119883 = 2119862 119905 = 119888 See Table 34
3144 119883 = 2119863 119905 = (119888(119888119889)12)11 See Table 35
315 119870 cong 1198621199003
3151 119883 = 2119860 119905 = 1198872 See Tables 36 and 37
12 Algebra
Table 39
119883119862
|O| 119892
8119862
23760 11988611988721198861198872
23760 1198872119886211988721198862
23760 1198872119886119887119886
23760 11988711988621198871198861198873
911986031680 119887119886119887
21198861198873
31680 119886119887119886119887119886119887119886119887
9119861 63360 1198861198871198861198871198861198872
1011986047520 119886119887119886
21198872
47520 1198871198861198871198861198872
1011986147520 119886119887119886
47520 119886211988731198862
1211986023760 119886
21198871198862
23760 1198861198873119886
12119861
47520 119887
47520 1198861198871198861198871198862
47520 1198871198861198872119886119887
1411986095040 119887119886119887119886
95040 1198861198872119886119887119886
1511986095040 119886119887
2
95040 119886119887119886119887
15119861 190080 1198861198871198861198872
1811986095040 119886119887
2119886119887
95040 119887119886119887119886119887
21119860 190080 11988711988621198871198862
2411986095040 119886119887
2119886
95040 119886211988721198862
2411986195040 119886
2119887119886
95040 119887119886
3011986095040 119886119887119886
2
95040 11988611988731198862
3152 119883 = 2119861 119905 = ((1198862119887)3119886119887)5 See Tables 38 and 39
316 119870 cong 1198621199002
3161 119883 = 2119860 119905 = 119886 See Table 40
3162 119883 = 2119861 119905 = 119886(119886119887119886119887119886)2119886(119886119887119886119887119886)7 See Table 41
3163 119883 = 2119862 119905 = (119887minus1(1198861198872)2119886119887minus1(1198861198872)4)5 See Table 42
Table 40
119883119862
|O| 119892
2119861 1260 1198871198861198872119886119887
2119862 1008 11988731198861198871198861198873
3119861 14336 119887119886119887
4119862 40320 119887
Table 41
119883119862
|O| 119892
2119860 15 (1198874119886)2119887311988611988721198861198874(119887119886119887)
2119886(119887119886119887)
711988611988711988611988731198861198872(119886119887)2
21198611680 119887
31198861198871198861198873
210 11988741198861198874119886119887119886119887
2119862 2520 (119886119887)5119886(119886119887)
2119886(119886119887)
3
3119861 35840 1198872119886119887
4119860 1920 11988711988611988731198861198872119886119887119886119887
4119862 20160 1198871198861198872119886119887
4119864 161280 119887119886119887
411986513440 119887119886119887119886119887119886119887119886119887
13440 119887311988611988711988611988721198861198872
5119861 344064 1198872
6119864 430080 119887
Table 42
119883119862
|O| 119892
2119860240 119886119887
4119886119887211988611988711988611988741198861198873119886119887311988611988741198861198873119886119887119886
45 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887411988611988741198861198872119886
2119861
720 1198872119886
240 11988741198861198874119886119887311988611988741198861198874119886119887211988611988731198861198872119886
180 119886119887211988611988731198861198872119886119887119886119887311988611988731198861198872
30 1198861198871198861198874119886
2119862
1440 1198861198871198861198873119886119887311988611988721198861198871198861198872119886
1440 119886119887311988611988741198861198873119886119887211988611988721198861198874119886
1440 1198872119886119887119886119887119886119887
311988611988741198861198873
1152 11988611988711988611988731198861198873119886119887311988611988741198861198872119886
360 11988611988711988611988711988611988731198861198872119886119887119886119887311988611988731198861198872
3119860 10240 1198861198874119886119887211988611988711988611988721198861198871198861198872119886
3119861
15360 11988611988731198861198872119886
7680 11988731198861198872
1024 1198861198872119886119887211988611988731198861198873119886
41198603840 119887
411988611988741198861198873119886119887411988611988741198861198874
1440 1198861198871198861198872119886119887211988611988741198861198872
4119861
11520 1198871198861198874119886119887411988611988731198861198873119886
5760 119886119887211988611988711988611988721198861198871198861198873119886
5760 11988611988721198861198874119886119887311988611988741198861198873119886
1440 11988611988741198861198871198861198872119886119887411988611988721198861198871198861198873119886
1440 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887119886
720 119886119887119886119887411988611988721198861198872119886119887211988611988731198861198873
720 1198872119886119887411988611988721198861198872119886119887411988611988731198861198874119886
Algebra 13
Table 42 Continued
119883119862
|O| 119892
4119862
11520 11988611988721198861198873119886
11520 1198861198873119886119887119886119887311988611988721198861198872
5760 119887311988611988741198861198872119886
5760 11988611988731198861198871198861198872
2880 11988611988711988611988741198861198872119886
1440 119887411988611988741198861198873119886119887211988611988721198861198873119886
1440 1198861198872119886119887311988611988731198861198872119886119887119886119887
720 11988611988711988611988711988611988711988611988721198861198871198861198873119886
720 119886119887311988611988741198861198872119886
4119863
23040 1198871198861198873119886119887119886119887211988611988731198861198872
11520 119886119887119886119887311988611988731198861198871198861198873
5760 119886119887411988611988711988611988741198861198872119886
2880 11988611988711988611988711988611988741198861198872119886119887411988611988741198861198873119886
2880 1198861198872119886119887119886119887119886119887
311988611988711988611988741198861198874119886
4119864
23040 1198871198861198871198861198871198861198872
23040 119886119887311988611988721198861198872
11520 1198861198873119886119887211988611988741198861198872
11520 119887311988611988711988611988731198861198872119886
5760 11988731198861198874119886
5760 1198861198871198861198872
5760 1198874119886119887311988611988721198861198871198861198873119886
4119865
23040 119886119887119886119887211988611988721198861198872
23040 1198873119886119887311988611988731198861198874119886
11520 119887411988611988741198861198872119886119887411988611988731198861198872119886
11520 119886119887311988611988741198861198873119886119887211988611988721198861198872
5760 1198872119886119887411988611988721198861198872119886119887211988611988731198861198872119886
5760 1198872119886119887211988611988731198861198873119886119887411988611988721198861198873119886
5760 1198861198873119886119887211988611988741198861198872119886119887311988611988731198861198872119886
5760 1198861198874119886119887119886119887119886119887
3119886119887211988611988721198861198873
5760 119886119887211988611988731198861198871198861198872119886119887211988611988731198861198873
5760 1198874119886119887411988611988721198861198872119886119887211988611988731198861198872119886
2880 119887311988611988721198861198871198861198872
2880 1198873119886119887411988611988731198861198872
1920 11988611988711988611988741198861198872119886119887211988611988741198861198873
1920 1198871198861198873119886119887311988611988711988611988721198861198874
4119866
46080 1198873119886119887311988611988721198861198873119886
46080 1198861198872119886119887311988611988721198861198872
23040 1198861198871198861198872119886119887211988611988731198861198872119886
23040 11988611988731198861198872119886119887311988611988731198861198874119886
5119860 73728 119886119887211988611988731198861198871198861198872119886
5119861184320 119887
21198861198872119886
122880 119887211988611988731198861198872
Table 42 Continued
119883119862
|O| 119892
6119860 92160 1198871198861198873119886119887411988611988731198861198872119886
6119861 122880 119887211988611988741198861198872119886
6119862
46080 1198872119886119887311988611988731198861198872119886
46080 119887311988611988711988611988721198861198872
15360 1198874119886119887211988611988731198861198873119886
15360 119886119887211988611988721198861198873119886119887
6119863
92160 119887411988611988711988611988721198861198874119886
23040 11988741198861198873119886119887411988611988721198861198872
15360 11988611988721198861198872119886119887119886
15360 119886119887411988611988731198861198873119886
6119864
92160 1198871198861198871198861198872
92160 1198871198861198874119886
46080 1198873119886119887211988611988731198861198872119886
46080 11988711988611988721198861198873119886
46080 11988721198861198871198861198874119886
46080 11988611988711988611988741198861198873
15360 119886119887211988611988711988611988741198861198871198861198872119886
6119865
92160 1198861198873119886119887119886
92160 11988611988741198861198872119886
46080 119887311988611988721198861198871198861198872119886
46080 1198861198874119886119887411988611988731198861198873
46080 1198861198873119886119887411988611988731198861198872
46080 11988721198861198872119886119887119886119887119886
7119860 737280 11988741198861198872
811986092160 119887
2119886119887119886
92160 11988611988741198861198873
8119861
92160 119886119887411988611988721198861198872119886119887119886
92160 119886119887411988611988731198861198873119886119887119886
46080 11988611988711988611988731198861198872119886
46080 119886119887311988611988721198861198874119886
23040 119887119886119887311988611988731198861198871198861198873
23040 11988721198861198874119886119887211988611988721198861198874
8119862
92160 1198873119886119887311988611988731198861198874
92160 1198872
92160 1198873
92160 119887119886119887211988611988721198861198872
8119863
92160 119887311988611988731198861198872
92160 11988611988721198861198873119886119887311988611988741198861198872
92160 119887411988611988731198861198873119886
92160 11988711988611988721198861198871198861198873119886
92160 119887311988611988721198861198872
14 Algebra
Table 42 Continued
119883119862
|O| 119892
8119864
92160 1198861198873119886
92160 1198861198872119886
46080 119886119887311988611988711988611988741198861198872
46080 119887311988611988711988611988741198861198872119886
46080 119886119887211988611988721198861198871198861198873
46080 1198872119886119887119886119887119886
46080 1198872119886119887411988611988731198861198873119886
46080 1198872119886119887211988611988731198861198873119886
46080 1198861198872119886119887211988611988731198861198873
46080 119886119887411988611988741198861198873
23040 1198861198874119886119887211988611988721198861198871198861198873
23040 11988741198861198874119886119887211988611988741198861198873
8119865
184320 11988611988741198861198871198861198873119886
184320 11988611988721198861198874119886119887119886
92160 119886119887119886119887311988611988711988611988721198861198872
92160 11988731198861198874119886119887411988611988731198861198872119886
9119860737280 119886119887
41198861198871198861198872
245760 119887211988611988741198861198872
10119860 368640 119887211988611988731198861198872119886
10119861368640 119886119887
21198861198872
368640 11988731198861198873119886
10119862368640 119886119887
4
368640 11988611988711988611988731198861198871198861198873119886
11119860 1474560 11988721198861198873119886
12119860122880 119887119886119887
21198861198871198861198874119886
122880 119887411988611988711988611988741198861198873
12119861368640 119887
211988611988711988611988731198861198872
184320 1198861198874119886119887211988611988741198861198872
12119862368640 119886119887
2119886119887211988611988731198861198872
184320 119886119887411988611988711988611988721198861198872
12119863
184320 119887411988611988711988611988721198861198872
184320 11988611988711988611988711988611988731198861198873119886
92160 119886119887411988611988721198861198873119886
92160 11988611988721198861198873119886119887119886
12119864368640 119887
311988611988721198861198874
368640 11988711988611988731198861198872
12119865
184320 11988721198861198871198861198872119886
184320 11988611988731198861198874119886
184320 11988611988711988611988731198861198873
184320 1198861198871198861198872119886
92160 1198861198872119886119887119886
92160 11988611988741198861198873119886
12119866 368640 11988711988611988741198861198872119886
Table 42 Continued
119883119862
|O| 119892
12119867
184320 1198874119886119887311988611988731198861198872
184320 119886119887311988611988721198861198873119886
184320 11988611988711988611988711988611988741198861198872119886
184320 1198861198874119886119887311988611988741198861198872119886
184320 119887311988611988721198861198872119886119887
184320 119886119887211988611988731198861198872119886
14119860 737280 1198861198874119886
15119860737280 119887
4119886
737280 1198871198861198872119886
16119860
368640 11988611988731198861198872
368640 11988611988721198861198871198861198874
368640 11988731198861198872119886
368640 11988711988611988741198861198873119886
18119860 737280 11988731198861198873
20119860737280 119886119887
41198861198872
737280 1198873119886119887119886
20119861737280 119886119887
41198861198874119886
737280 119886119887119886119887119886
24119860737280 119886119887
2
737280 1198873119886
24119861737280 119887
4
737280 119887
28119860737280 119886119887119886119887
31198861198872
737280 119886119887411988611988721198861198872
30119860737280 119887
21198861198874
737280 1198871198861198873
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R H Bing ldquoA homeomorphism between the 3-sphere andthe sum of two solid horned spheresrdquo Annals of MathematicsSecond Series vol 56 pp 354ndash362 1952
[2] D Montgomery and L Zippin ldquoExamples of transformationgroupsrdquo Proceedings of the American Mathematical Society vol5 pp 460ndash465 1954
[3] J W Morgan and H Bass Eds The Smith Conjecture vol 112of Pure and Applied Mathematics Academic Press OrlandoFla USA 1984 Papers presented at the symposium held atColumbia University New York NY USA 1979
[4] C M Edwards and G T Ruttimann ldquoInvolutive and Peircegradings in JBWlowast-triplesrdquo Communications in Algebra vol 31no 6 pp 2819ndash2848 2003
Algebra 15
[5] J Gornicki and B E Rhoades ldquoA general fixed point theoremfor involutionsrdquo Indian Journal of Pure and Applied Mathemat-ics vol 27 no 1 pp 13ndash23 1996
[6] S Montgomery Fixed Rings of Finite Automorphism Groupsof Associative Rings vol 818 of Lecture Notes in MathematicsSpringer Berlin Germany 1980
[7] T A Springer ldquoThe classification of involutions of simplealgebraic groupsrdquo Journal of the Faculty of Science University ofTokyo Section IA Mathematics vol 34 no 3 pp 655ndash670 1987
[8] T A Springer ldquoSome results on algebraic group s with invo-lutionsrdquo in Algebraic Groups and Related Topics (KyotoNagoya1983) vol 6 of Advanced Studies in Pure Mathematics pp 525ndash543 North-Holland Amsterdam The Netherlands 1985
[9] W Feit and J GThompson ldquoSolvability of groups of odd orderrdquoPacific Journal of Mathematics vol 13 pp 775ndash1029 1963
[10] R Brauer and K A Fowler ldquoOn groups of even orderrdquo Annalsof Mathematics vol 62 no 3 pp 565ndash583 1955
[11] D Gorenstein The Classification of Finite Ssimple Groups Vol1 Groups of Noncharacteristic 2 Type The University Series inMathematics Plenum Press New York NY USA 1983
[12] D Gorenstein R Lyons and R Solomon The Classification ofthe Finite Simple Groups vol 40 of Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 1994
[13] C Bates and P Rowley ldquoInvolutions in Conwayrsquos largest simplegrouprdquo LMS Journal of Computation andMathematics vol 7 pp337ndash351 2004
[14] P Rowley and P Taylor ldquoPoint-line collinearity graphs of twosporadic minimal parabolic geometriesrdquo Journal of Algebra vol331 pp 304ndash310 2011
[15] P Rowley and P Taylor ldquoInvolutions in Jankorsquos simple group 1198694rdquoLMS Journal of Computation and Mathematics vol 14 pp 238ndash253 2011
[16] P Taylor ldquoInvolutions in Fischerrsquos sporadic groupsrdquo preprinthttpeprintsmamanacuk1622
[17] The GAP Group GAP-Groups Algorithms Programming Ver-sion 43 2002 httpwwwgap-systemorg
[18] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I The user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997
[19] R Wilson P Walsh J Tripp et al ldquoAtlas o f finite grouprepresentationsrdquo httpbrauermathsqmulacukAtlasv3
[20] J H Conway R T Curtis S P Norton R A Parker and R AWilson Atlas of Finite Groups Clarendon Oxford UK 1985
[21] R AWilson ldquoStandard generators for sporadic simple groupsrdquoJournal of Algebra vol 184 no 2 pp 505ndash515 1996
[22] D Gorenstein Finite Groups Chelsea Publishing New YorkNY USA 2nd edition 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Algebra
Fltabgt=FreeGroup(2)
phi=homlt F-gtG | ltaG1gtltbG2gt gt
rh=IsConjugate(Gtx)
sh=hphi
lengthmin=sh
hmin=h
Ct=Centralizer(Gt)
for c in Ct do
sch=(clowasth)phi
if sch lt lengthmin then
lengthmin=(sch)
hmin=clowasth shmin=sch
end if
end for
Algorithm 1
wemay write our conjugating element 119892119894as a word in 119886 and 119887
Thus our aim is to find 119892119894which have relatively ldquosmallrdquo length
relative to the generating set 119886 119887 for119870 However wemay notalways achieve the minimum possible length In more detailwe proceed as follows define119863
0(119886 119887) = 1 and for 119895 isin N
119863119895(119886 119887) = 119909119886 119909119887 | 119909 isin 119863
(119895minus1)(119886 119887) (1)
For 119899 isin N set
119863[119899](119886 119887) = ⋃
119895le119899
119863119895(119886 119887) (2)
Themain purpose of119863[119899](119886 119887) is to produce a colony of short
words in 119886 and 119887 a number of which may well yield thesame element of 119866 To speed matters up we prune out theseduplicates We hunt through119863
[119899](119886 119887) for 119899 typically at most
20 so as to ensnare suitable conjugating elements 119892119894 If this
does not yield enough representatives for the 119862119866(119905)-orbits
then we recalculate 119863[119899](119886 119887) with 119886 and 119887 replaced by short
words in 119886 and 119887which are also generators for119866 For examplewe might try replacing 119886 by 119886119887 and keeping 119887 the same ortry a more complicated substitution such as replacing 119887 with(119886119887)5119886 Such a substitution was used to produce a 119892
119894in the
case when 119866 cong 11987224
and 119905119909 in the class 3119860 starting fromthe word 11988711988611988721198861198871198861198872119886119887 Some of the words for the 119892
119894may be
further simplified and this was done by handOn a number of occasions this approach fails to deliver
conjugating elements for some (often of small size) 119862119866(119905)-
orbits To deal with such elusive119862119866(119905)-orbits sayO we begin
by finding an 119909 in O usually by a random search Then weobtain an element ℎ in 119866 such that 119905ℎ = 119909 Now we runthrough some or all of the elements 119888 in 119862
119866(119905) seeking a 119888 for
which 119888ℎ has small ldquolengthrdquo (ie the total number of symbols119886 119886minus1 119887 119887minus1 in the given expression for 119888ℎ) Sometimes we
may also vary ℎ In somewhat more detail the Magma codeemployed is as shown in Algorithm 1
We remark that this procedure is not usually as efficientas using the set 119863
[119899](119886 119887) and moreover usually yields more
complicated expressions for the conjugating elementsWe say a few words about how we guarantee that
we have a representative of every 119862119866(119905) orbit By using
the IsConjugate(Ctxy) command where 119905 is a fixedinvolution of the 119866-conjugacy class and Ct denotes 119862
119866(119905)
we ensure that no two representatives are in the same 119862119866(119905)-
orbitWe terminate our calculations for the119866-conjugacy classof 119905 when the sum of the 119862
119866(119905)-orbit sizes equals the size of
the conjugacy classOne may envision the information presented here to be
useful in the following circumstances Suppose 119867 is a finitegroup with 119909 119910 isin 119867 Further suppose that we have identifiedthe subgroup 119871 = ⟨119909 119910⟩ as being isomorphic to 119866 viarecognizing 119909 and 119910 as standard generators (See [21] for adiscussion of standard generators)Thenwemay translate ourinformation from 119866 to 119871 so that we see how 119871 acts upon itsinvolution conjugacy classes within the group119867 Moreover if119867 is large from a computational standpoint (eg |119867| is largeor119867 is a matrix group of large degree) then having the 119892
119894as
short words may be beneficial To facilitate applications suchas these computer files containing the 119892
119894as words in 119886 and 119887
are available on request from the second authorThe following section gives the permutation ranks of 119866
on 119883 and the elements 119892119894together with some additional
information
3 Orbit Representatives
So 119905 is a suitably chosen (and then fixed) involution of theconjugacy class119883 of 119866 For 119862 a 119866-conjugacy class we define
119883119862= 119909 isin 119883 | 119905119909 isin 119862 (3)
Since 119883119862is 119862119866(119905)-invariant 119883
119862will be a union of 119862
119866(119905)-
orbits for each 119866-conjugacy class 119862 for which 119883119862
= 0 As iswell known |119883
119862|may be calculated for any119866-conjugacy class
using the complex character table (see eg [22]) and this iseasily carried out in Gap As we proceed by breaking each119883
119862
(for |119883119862| = 0) into 119862
119866(119905)-orbits this is useful information
Our first table gives an overview of the permutation ranksof119866 in its action on119883The succeeding tables consider in turnthe possibilities for 119870 with the first column identifying 119883
119862
and the second the size of the119862119866(119905)-orbitO (contained in119883
119862)
and the third supplies a group element 119892 for which 119905119892 isin OWe emphasize that the following tables give the 119862
119870(119905)-
orbits of 119883mdashin the case when 119870 lt 119866 = Aut(119870) the tablesare annotated so as to also yield the 119862
119866(119905)-orbits of119883 Before
explaining how this is done we remark that in all instanceshere when 119870 lt 119866 = Aut(119870) we have 119862
119870(119905) lt 119862
119866(119905) Thus
119883 is a 119866-conjugacy class and 119866 = 119870119862119866(119905) Now suppose that
1198621and 119862
2are two 119870-conjugacy classes which fuse in 119866 (so
1198621cup 1198622is a 119866-conjugacy class) Then 119862
119892
1= 1198622for some
119892 isin 119862119866(119905) and consequently 119883119892
1198621
= 1198831198622
Hence if O is a119862119870(119905)-orbit contained in 119883
1198621
O cup O119892 will be a 119862119866(119905)-orbit
of 119883 In this circumstance a broken horizontal line indicatesthat 119862
1and 119862
2are fused in 119866 = Aut(119870) and if O is obtained
by using the 119894th listed conjugating element in 1198831198621
then O119892
is obtained using the 119894th listed conjugating element in 1198831198622
When a 119870-conjugacy class 119862 is also a 119866-conjugacy class itmay be the case that a 119862
119866(119905)-orbit contained in 119883
119862is the
union of two 119862119870(119905)-orbits in 119883
119862 A vertical line connecting
Algebra 3
Table 1
119866 2119860 2119861 2119862 2119863
11987211
811987212
10 11 131198691
2211987222
13 5 101198692
6 31 1411987223
9119867119878 9 27 5 321198693
27 1711987224
14 26119872119888119871 6 28119867119890 12 34 41119877119906 22 42119878119906119911 9 66 15 411198621199003
7 611198621199002
5 13 178
Table 2
119883119862
|O| 119892
2119860 12 1198871198861198873119886119887
311986024 119887
2
8 1198873119886119887
4119860 24 119887119886119887
5119860 48 119887119886119887119886119887
611986024 119887
24 1198873
Table 3
119883119862
|O| 119892
211986010
10
1003816100381610038161003816100381610038161003816100381610038161003816
11988721198861198871198861198872119886119887119886119887
119887211988611988721198861198871198861198872119886119887
2119861 15 119886
3119861 60 119886119887
4119860 30 1198861198872119886119887119886
4119861 30 1198871198861198871198861198871198861198872119886
5119860 120 119887119886
611986060
60
1003816100381610038161003816100381610038161003816100381610038161003816
119887
1198872
two 119862119870(119905)-orbits in 119883
119862signifies that their union is a 119862
119866(119905)-
orbit
31 Permutation Ranks See Table 1
32 119870 cong 11987211
321119883 = 2119860 119905 = 119886 See Table 2
33 119870 cong 11987212
331 119883 = 2119860 119905 = ((119886119887)21198861198872)3 See Table 3
Table 4
119883119862
|O| 119892
211986124 119887119886119887
2119886119887
6 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886119887119886119887
311986032
32
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887
1198871198861198872
3119861 16 11988711988611988721198861198872119886119887119886119887
4119860 48 1198871198861198871198861198872
4119861 48 1198872119886119887119886119887
5119860 96 119887119886119887
611986196
96
1003816100381610038161003816100381610038161003816100381610038161003816
119887
1198872
Table 5
119883119862
|O| 119892
211986015 119887119886119887
2119886119887119886119887119886119887
1 119887119886119887119886119887119886119887211988611988721198861198871198861198872119886119887
2119861 15 1198871198861198871198861198872
311986020
20
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887
1198861198871198861198872119886119887119886
3119861 60 1198871198861198872
511986060 119886119887119886119887
2119886
60 119886119887
6119860 60 119887119886119887119886
611986160
60
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887
119886
1011986060 119887119886119887
2119886119887119886
60 119887119886119887119886119887
11119860 120 119887
11119861 120 119887119886
332 119883 = 2119861 119905 = 119886 See Table 4
333 119883 = 2119862 119905 = 119888 See Table 5
34 119870 cong 1198691
341 119883 = 2119860 119905 = 119886 See Table 6
35 119870 cong 11987222
351 119883 = 2119860 119905 = 119886 See Table 7
352 119883 = 2119861 119905 = 119888 See Table 8
353 119883 = 2119862 119905 = (1198881198891198881198892)5 See Table 9
36 119870 cong 1198692
361 119883 = 2119860 119905 = (1198871198861198872(119886119887)2119887(119886119887)3)5 See Table 10
4 Algebra
Table 6
119883119862
|O| 119892
211986015 119887119886119887
2119886119887119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
15 119887119886119887211988611988721198861198871198861198872119886119887119886119887119886119887
2119886119887119886119887
311986060 119887119886119887
2119886119887119886119887119886119887
2119886119887
12 1198872119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
511986060 119887119886119887119886119887
21198861198872119886119887119886119887
20 1198872119886119887119886119887119886119887
21198861198872119886119887
511986160 119887119886119887119886119887
21198861198871198861198872119886119887119886119887
20 11988721198861198871198861198872119886119887119886119887
611986060 119887119886119887
2119886119887119886119887
60 1198871198861198871198861198872119886119887
7119860 120 11988711988611988721198861198872119886119887119886119887
1011986060 119887
2119886119887119886119887119886119887
60 1198871198861198871198861198871198861198872
1011986160 119887
2119886119887119886119887
60 1198871198861198871198861198872
11119860 120 1198871198861198871198861198872119886119887119886119887
15119860 120 11988721198861198872119886119887119886119887
15119861 120 119887119886119887
19119860 120 119887
19119861 120 1198872119886119887
19119862 120 11988721198861198871198861198872119886119887
Table 7
119883119862
|O| 119892
2119860
24 1198871198861198873119886119887
12 (119887119886)21198872119886119887119886119887
8 11988721198861198871198861198872(119886119887)3
6 1198873(119886119887)2(1198861198873)2119886119887
3119860 192 119887119886119887
4119860
48 1198871198861198872119886119887
48 1198873119886119887119886119887
48 1198871198861198871198861198873
411986196 119887
2119886119887119886119887
96 1198871198861198871198861198872
5119860 384 1198872
6119860 192 119887
Table 8
119883119862
|O| 119892
211986042 119887
7 (1198872119886)2119887119886
3119860 112 119886
4119860 168 119886119887119886119887119886
Table 9
119883119862
|O| 119892
211986020 119887
2
5 11988711988611988721198861198873
3119860 80 119887
4119860 80 119887119886119887119886119887119886
411986140 119887119886119887
2119886
40 1198871198861198872119886119887
5119860 320 119886119887
611986080
80
1003816100381610038161003816100381610038161003816100381610038161003816
119886
1198861198872119886
11119860 320 1198872119886
11119861 320 119887119886
Table 10
119883119862
|O| 119892
2119860 10 119887119886
3119861 160 119887
4119860 80 119886
5119860 32 1198861198872119886119887119886
5119861 32 1198871198861198872
362 119883 = 2119861 119905 = 119886 See Table 11
363 119883 = 2119862 119905 = 119888 See Table 12
37 119870 cong 11987223
371119883 = 2119860 119905 = 119886 See Table 13
38 119870 cong 119867119878
381 119883 = 2119860 119905 = 119886 See Table 14
382 119883 = 2119861 119905 = ((119886119887)2119886)5 See Tables 15 and 16
383 119883 = 2119862 119905 = 119888 See Table 17
384 119883 = 2119863 119905 = ((119889119888)21198894119888119889)5 See Tables 18 and 19
39 119870 cong 1198693
391119883 = 2119860 119905 = 119886 See Table 20
392119883 = 2119861 119905 = 119888 See Table 21
310 119870 cong11987224
3101 119883 = 2119860 119905 = ((119886119887)31198861198872)4 See Tables 22 and 23
Algebra 5
Table 11
119883119862
|O| 119892
2119860 15 119887(119887119886)2119887(119887119886)
2119887(119887119886)
2
2119861
15
15
1003816100381610038161003816100381610038161003816100381610038161003816
((119887119886)2119887)3(119887119886)3
119887(119887119886)2(119887(119887119886)
31198872119886)2
1
1
1003816100381610038161003816100381610038161003816100381610038161003816
(119887119886)21198872119886(119887(119887119886)
2)21198872119886119887(119887119886)
3
1198872119886119887(119887119886)
3119887(119887119886)
2119887(119887119886)
3(1198872119886)2
3119860 24 119887(119887119886)2119887(119887119886)
2
3119861 60 119887(119887119886)3119887(119887119886)
2119887(119887119886)
3119887(119887119886)
4119860 60 (119887119886)119887(119887119886)2
5119860 60 (119887119886)2119887(119887119886)
2119887(119887119886)
3119887(119887119886)119887(119887119886)
2
5119861 60 (119887119886)119887(119887119886)119887(119887119886)3119887(119887119886)
2119887(119887119886)
3
5119862120 119887(119887119886)
3119887(119887119886)
3
24 119887(119887119886)119887(119887119886)2119887(119887119886)
3
5119863120 (119887119886)119887(119887119886)
2119887(119887119886)
3
24 (119887119886)2119887(119887119886)
2119887(119887119886)119887(119887119886)
6119860 120 (119887119886)3
611986160
60
1003816100381610038161003816100381610038161003816100381610038161003816
(119887119886)2119887(119887119886)
3
(119887119886)3119887(119887119886)
4
7119860 240 119887(119887119886)4
1011986060 119887(119887119886)119887(119887119886)
2119887(119887119886)
2
60 119887(119887119886)2119887(119887119886)
3
1011986160 (119887119886)
2119887(119887119886)
2119887(119887119886)
60 (119887119886)119887(119887119886)2119887(119887119886)119887(119887119886)
10119862120 (119887119886)119887(119887119886)119887(119887119886)
120 (119887119886)2119887(119887119886)
10119863120 119887(119887119886)
3
120 119887(119887119886)119887(119887119886)2
12119860120
120
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119887119886)
(119887119886)
15119860 240 (119887119886)119887(119887119886)119887(119887119886)2119887(119887119886)
2
15119861 240 (119887119886)119887(119887119886)2119887(119887119886)119887(119887119886)
2
Table 12
119883119862
|O| 119892
2119860 21 (1198871198861198872119886)2(119887119886)3
2119861 28 1198871198861198872
311986014
14
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)3(1198861198872)2119886119887
(1198872119886119887119886)2119887119886119887
3119861 168 119887119886119887119886
411986084 119886119887119886119887
2119886119887
42 1198872119886119887119886119887
6119860
84
84
1003816100381610038161003816100381610038161003816100381610038161003816
1198872
119887
42
42
1003816100381610038161003816100381610038161003816100381610038161003816
119886119887119886119887119886119887119886
1198861198871198861198872119886119887119886119887
6119861 168 119886119887
7119860 336 119887119886119887
8119860168 119886119887
2119886
168 119886119887119886
12119860168
168
1003816100381610038161003816100381610038161003816100381610038161003816
119887119886
119886
Table 13
119883119862
|O| 119892
211986084 119887119886119887
3119886119887
14 11988731198861198872119886119887
3119860 448 1198872
4119860
336 119887119886119887119886119887
336 119887119886119887119886119887119886119887
336 1198871198861198871198861198871198861198872
5119860 896 119887119886119887
6119860 1344 119887
Table 14
119883119862
|O| 119892
211986080 119887
211988611988731198861198872
30 119887211988611988711988611988721198861198871198861198872
3119860 640 119887119886119887
4119861 480 1198871198861198873119886119887
4119862 960 1198872119886119887
5119861 128 1198871198861198872119886119887
5119862 1536 119887
6119861 1920 1198872
Table 15
119883119862
|O| 119892
211986045 119887119886119887
3119886119887
30 11988741198861198874119886119887119886119887
211986136
36
1003816100381610038161003816100381610038161003816100381610038161003816
119887211988611988731198861198872
1198871198861198871198861198871198861198873119886
3119860360 119887
31198861198872119886119887119886119887
24 1198861198872119886119887211988611988711988611988721198861198872119886
4119861 180 119886
4119862
360 11988711988611988711988611988741198861198872119886
180
180
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198871198861198872
1198872119886119887119886119887119886119887
5119860144
144
1003816100381610038161003816100381610038161003816100381610038161003816
11988711988611988741198861198873119886119887
11988711988611988731198861198874119886119887
5119861 720 1198871198861198871198861198872119886
Table 16
119883119862
|O| 119892
6119861
720 119887119886119887
360 1198871198861198872119886
360 1198861198872119886119887
7119860 2880 119887
8119861720 119887
41198861198872119886
720 1198861198873119886119887
8119862720 119887119886119887
3119886
720 11988611988721198861198874
10119860720
720
1003816100381610038161003816100381610038161003816100381610038161003816
1198872
1198873
10119861720
720
1003816100381610038161003816100381610038161003816100381610038161003816
119886119887119886
119887119886
15119860 2880 1198872119886
6 Algebra
Table 17
119883119862
|O| 119892
2119860 105 119886
2119861 28 119887119886
3119860 336 1198872
4119861 630 119887
Table 18
119883119862
|O| 119892
211986060 119886119887
31198861198871198861198872119886
5 11988611988731198861198872119886
2119861 60 1198872119886
3119860240 119886
160 11988611988731198861198871198861198872119886119887119886
4119860 120 1198871198861198873119886119887119886
4119861
240 119886119887
120 1198871198861198872119886119887119886119887
30 1198861198872119886
4119862120 119886119887
21198861198872119886
120 1198861198873119886119887119886
5119860192 119886119887119886119887
3
192 11988711988611988721198861198873
5119861 320 1198874119886119887
5119862 1920 11988721198861198872
6119860480 119886119887
2119886119887
480 1198871198861198872119886
Table 19
119883119862
|O| 119892
6119861480 119887
4119886119887119886119887
480 11988721198861198872119886
7119860 1920 119887
8119860480 119887119886119887119886119887
2
480 11988711988611988721198861198872119886119887
10119860960 119887
211988611988721198861198872119886
960 119887119886119887
10119861 960 1198861198872
11119860 1920 1198872119886119887119886
11119861 1920 119887119886119887119886119887
12119860960 119887119886
960 1198873
15119860 1920 1198871198861198873
20119860960 119887119886119887119886
960 1198872119886119887
20119861960 119887
21198861198873119886
960 1198874119886119887119886
Table 20
119883119862
|O| 119892
2119860120 119887(119887119886)
3119887(119887119886)119887(119887119886)
5
10 (119887119886)119887(119887119886)4119887(119887119886)
2
3119860 192 119887(119887119886)6
3119861 640 (119887119886)119887(119887119886)2
4119860480 119887(119887119886)
5119887(119887119886)
80 (119887119886)3119887(119887119886)
3119887(119887119886)
4
5119860960 (119887119886)
3119887(119887119886)
3119887(119887119886)
2
320 (119887119886)2119887(119887119886)
2
5119861960 (119887119886)119887(119887119886)
3119887(119887119886)
4
320 (119887119886)119887(119887119886)3
6119860 960 (119887119886)119887(119887119886)6
8119860960 (119887119886)
4119887(119887119886)
2
960 (119887119886)119887(119887119886)5
9119860 1920 119887(119887119886)2119887(119887119886)
2
9119861 1920 (119887119886)
9119862 1920 119887(119887119886)2
10119860960 (119887119886)
4119887(119887119886)
960 119887(119887119886)4119887(119887119886)119887(119887119886)
10119861960 119887(119887119886)119887(119887119886)119887(119887119886)119887(119887119886)
2
960 119887(119887119886)5
12119860960
960
1003816100381610038161003816100381610038161003816100381610038161003816
(119887119886)2119887(119887119886)
4
(119887119886)6
15119860 1920 (119887119886)2119887(119887119886)
15119861 1920 119887(119887119886)3
17119860 1920 (119887119886)2
17119861 1920 (119887119886)2119887(119887119886)
3
Table 21
119883119862
|O| 119892
2119860 153 1198861198872119886119887119886119887119886119887
3119860102
102
1003816100381610038161003816100381610038161003816100381610038161003816
1198872
119887
3119861 272 1198861198871198861198872119886
4119860612 119887119886119887119886119887119886119887
2119886
306 1198871198861198872119886119887119886119887119886
6119860
612
612
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198872119886119887119886
1198871198861198871198861198872119886
306
306
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198872119886119887119886119887119886
119887119886119887119886
81198601224 119886119887119886119887119886119887
1224 1198872119886119887119886119887119886
9119860 816 11988721198861198871198861198872119886119887
9119861 816 1198871198861198871198861198872119886119887
9119862 816 1198871198861198872
121198601224
1224
1003816100381610038161003816100381610038161003816100381610038161003816
119886119887
1198861198872119886119887119886
17119860 2448 119887119886
17119861 2448 119886
19119860 2448 1198861198872119886119887
19119861 2448 1198872119886119887119886
Algebra 7
Table 22
119883119862
|O| 119892
2119860
168 1198872119886119887119886119887119886119887
2119886119887
14 119886119887119886
14 1198861198871198861198872119886119887
2119861 84 1198872119886119887
3119860 896 119887
3119861 128 119887211988611988721198861198871198861198872119886119887119886119887
Table 23
119883119862
|O| 119892
4119860112 119887119886119887119886119887119886119887
112 1198872119886119887211988611988721198861198872
4119861
1344 1198861198872119886119887119886
672 11988611988721198861198871198861198872119886119887
672 11988721198861198871198861198872119886119887119886
5119860 1792 119887119886119887119886
6119860 5376 119886
Table 24
119883119862
|O| 119892
211986060 119887
2119886119887119886119887211988611988721198861198871198861198872119886119887119886119887119886119887
2119886119887
15 11988711988611988721198861198872119886119887119886119887119886119887
2
2119861
120 119887119886119887211988611988711988611988721198861198871198861198872
80 119887211988611988711988611988721198861198871198861198872119886119887
2 (119887119886)8(1198872119886)21198871198861198872(119886119887)2(1198861198872)2(119886119887)3(1198861198872119886119887)2119887
3119860 320 119887211988611988721198861198871198861198872119886119887119886119887119886119887
3119861 960 11988721198861198871198861198872119886119887
4119860120 119887119886119887
21198861198872
120 1198871198861198871198861198872
4119861
480 1198872119886119887119886119887119886119887
2119886119887
240 1198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
240 1198872119886119887119886119887119886119887119886119887119886119887119886119887
4119862
960 1198871198861198872119886119887
960 11988721198861198871198861198872
160 1198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
21198861198872
160 1198872119886119887119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
5119860 1920 119887119886119887
6119860 1920 1198871198861198871198861198871198861198872
61198611920 119887119886119887
2
1920 1198872119886119887
101198601920 119887
21198861198871198861198872119886119887119886119887
1920 1198872119886119887119886119887119886119887119886119887
2
11119860 7680 119887119886119887119886119887
121198613840 119887
3840 1198872
3102 119883 = 2119861 119905 = 119886 See Table 24
311 119870 cong 119872119888119871
3111119883 = 2119860 119905 = 119886 See Table 25
3112119883 = 2119861 119905 = 119888 See Table 26
Table 25
119883119862
|O| 119892
2119860 210 119887211988611988731198861198872
3119861 2240 1198871198861198871198861198872
4119860 5040 1198872119886119887
5119861 8064 119887
6119861 6720 1198872
Table 26
119883119862
|O| 119892
2119860 165 11988611988721198861198871198861198872119886119887
3119860 220 1198872119886119887311988611988721198861198871198861198872119886
3119861660 119886119887
21198861198872
220 1198872119886119887119886119887119886119887
3
41198601980 119887
21198861198873119886119887119886119887119886
990 119886
5119860792 119887
211988611988721198861198873
792 1198872119886119887119886119887119886119887
2
5119861 7920 1198872
6119860 1980 119886119887
6119861
1980 1198871198861198872119886119887119886119887
1980 1198861198872119886119887
660
660
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198873
11988711988611988731198861198874119886119887
71198603960 119887119886119887119886119887
2119886
3960 1198871198861198872
71198613960 119887119886119887
3119886
3960 1198872119886
81198603960 119887
21198861198872119886
3960 11988611988731198861198873
9119860 2640 1198872119886119887119886
9119861 2640 1198861198871198861198872119886119887
101198603960 119887119886119887119886119887
3960 119886119887119886119887119886
121198603960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198873
1198871198861198873
141198603960 119886119887
21198861198873
3960 1198871198861198872119886119887
141198613960 119887119886119887119886119887119886
3960 1198872119886119887119886119887119886
151198603960 119887119886119887
3960 1198874
151198613960 119886119887119886119887
3960 119887
301198603960 119887119886
3960 1198871198861198872119886
301198613960 119887
41198861198872119886
3960 1198861198872
8 Algebra
Table 27
119883119862
|O| 119892
2119860240 119887119886119887119886119887119886119887
2 1198875(1198861198876)2(119886119887)31198875119886(11988721198861198873)2
2119861 315 119887211988611988711988611988721198861198872119886119887
3119860 1344 1198872119886119887
3119861 960 1198871198861198874119886119887
4119860480 119887
31198861198871198861198872119886119887
480 11988721198861198871198861198873119886119887
41198612520
2520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887
1198874119886119887
5119860 2688 1198873
6119860 13440 119887
Table 28
119883119862
|O| 119892
2119860 84 (119886119887)21198875(119886119887)2
2119861
168
168
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)2
(119886119887)1198874(119886119887)119887
2(119886119887)119887
2(119886119887)119887
4
14
14
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198874(119886119887)119887
4(119886119887)119887
2(119886119887)119887
2
1198872(119886119887)119887
2(119886119887)21198874(119886119887)119887
2(119886119887)119887
2
3119860 896 1198874(119886119887)119887
2
3119861 2688 1198872
4119861
1344
1344
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)119887
2(119886119887)119887
2
1198872(119886119887)119887
4(119886119887)21198874
112
112
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)1198872(119886119887)41198874(119886119887)2
(119886119887)31198874(119886119887)41198872
4119862
1344
1344
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)1198874(119886119887)119887
4
1198874(119886119887)119887
2(119886119887)119887
2(119886119887)31198872
672
672
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198872(119886119887)3
1198872(119886119887)41198872(119886119887)2
672
672
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)41198872(119886119887)2
(119886119887)4119887(119886119887)
2
5119860 5376 (119886119887)3
6119860 5376 (119886119887)1198872(119886119887)119887
2(119886119887)4
61198615376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198872(119886119887)
(119886119887)1198872(119886119887)
7119862 3072 1198872(119886119887)119887
2(119886119887)119887
2(119886119887)2
81198605376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)21198872(119886119887)21198872
(119886119887)1198872(119886119887)21198872(119886119887)
101198605376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)2
1198872(119886119887)119887
2(119886119887)119887
2
1211986110752
10752
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)2
1198874(119886119887)2
15119860 21504 1198872(119886119887)3
17119860 21504 (119886119887)1198872(119886119887)2
17119861 21504 (119886119887)
21119860 21504 (119886119887)1198872(119886119887)119887
2
21119861 21504 1198872(119886119887)
Table 29
119883119862
|O| 119892
2119860105 (119886119887)119887(119886119887)
3119887(119886119887)
3
63 119887611988611988731198861198873119886119887211988611988761198861198872119886119887611988611988761198861198874119886
2119861 315 (1198861198872)2(119887119886)21198872119886
3119860630 (119886119887)119887(119886119887)119887(119886119887)119887
126 119887(119886119887)21198876
3119861 2520 (119886119887)1198876
41198601260 (119886119887)
21198875(119886119887)
1260 (119886119887)119887(119886119887)1198873
4119861
945
945
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)21198872(119886119887)119887
119887(119886119887)119887(119886119887)1198872(119886119887)119887
315
315
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119886119887)5119887
119887(119886119887)31198872(119886119887)119887
41198621890
1890
1003816100381610038161003816100381610038161003816100381610038161003816
1198873(119886119887)2
(119886119887)1198872(119886119887)2119887
5119860 2520 (119886119887)3119887(119886119887)
6119860
3780 (119886119887)31198872(119886119887)2
1890 119887(119886119887)119887(119886119887)
1890 119887(119886119887)119887(119886119887)119887(119886119887)3
1260 119887(119886119887)1198872(119886119887)119887(119886119887)
2
6119861 7560 (119886119887)
71198601890 (119886119887)119887(119886119887)
41198872
90 119887(119886119887)1198873(119886119887)2119887
71198611890 119887
5(119886119887)119887(119886119887)
90 1198875(119886119887)119887
5(119886119887)119887
2(119886119887)119887
5
7119862 2160 119887(119886119887)119887(119886119887)4
71198637560 (119886119887)119887(119886119887)
4
1080 (119886119887)1198873(119886119887)
71198647560 (119886119887)
3119887(119886119887)
2
1080 (119886119887)1198874(119886119887)2119887(119886119887)
8119860
3780
3780
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)2119887(119886119887)
2
(119886119887)4
3780
3780
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)3119887(119886119887)119887(119886119887)119887
(119886119887)119887(119886119887)119887(119886119887)2
10119860 7560 (119886119887)3
121198607560 119887(119886119887)119887
7560 119887(119886119887)1198872
121198617560
7560
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119886119887)2119887
119887
14119860
3780 (119886119887)1198873(119886119887)2
3780 (119886119887)21198872(119886119887)
1890 (119886119887)1198872(119886119887)
1890 119887(119886119887)3119887
14119861
3780 (119886119887)119887(119886119887)2119887(119886119887)
3780 (119886119887)1198874
1890 (119886119887)21198872(119886119887)119887(119886119887)
2
1890 119887(119886119887)119887(119886119887)2
Algebra 9
Table 29 Continued
119883119862
|O| 119892
141198627560 119887(119886119887)
7560 119887(119886119887)3
141198637560 119887
3(119886119887)
7560 119887(119886119887)2
15119860 15120 (119886119887)1198872
21119860 15120 (119886119887)4119887(119886119887)
21119861 15120 (119886119887)119887
21119862 15120 (119886119887)3119887
21119863 15120 1198872
281198607560 119887
2(119886119887)119887
2
7560 (119886119887)2
281198617560 119887
4(119886119887)3
7560 (119886119887)119887(119886119887)2
Table 30
119883119862|O| 119892
2119860
640 11988611988731198861198872119886119887119886
480 11988611988721198861198872119886119887119886119887119886119887119886119887119886119887
21198861198872119886
240 11988611988721198861198873119886119887311988611988731198861198873119886119887211988611988711988611988721198861198873119886119887119886119887119886119887119886119887
31198861198873
30 1198861198872119886119887119886119887119886119887211988611988721198861198873119886119887311988611988711988611988711988611988721198861198873119886119887211988611988721198861198873119886119887211988611988711988611988711988611988731198863119860 10240 119886119887
3119886119887119886119887119886119887119886119887119886119887
2119886
4119860 5120 119886119887119886119887119886119887119886
4119862
7680 119886119887119886119887211988611988721198861198872119886119887119886
3840 119886119887211988611988731198861198871198861198872119886119887119886
3840 119886119887119886119887211988611988711988611988731198861198872119886
411986315360 119886119887119886
7680 1198861198872119886119887211988611988711988611988721198861198871198861198872119886
5119860 12288 11988611988731198861198871198861198872119886119887119886119887119886119887119886
5119861 4096 119886119887119886119887119886
611986061440 119886119887
21198861198871198861198872119886119887119886
30720 119886119887119886119887119886119887119886119887119886119887119886
7119860 61440 119886
811986261440 119886119887119886119887
21198861198872119886
61440 11988611988721198861198872119886119887119886
10119860 61440 11988611988721198861198872119886119887119886119887119886119887119886
12119860122880 1198861198872119886119887119886
13119860 61440 1198861198873119886119887119886
312 119870 cong 119867119890
3121119883 = 2119860 119905 = 119886 See Table 27
3122119883 = 2119861 119905 = (1198861198872)6 See Table 28
3123 119883 = 2119862 119905 = 1198893 See Table 29
313 119870 cong 119877119906
3131 119883 = 2119860 119905 = 1198872 See Table 30
3132 119883 = 2119861 119905 = 119886 See Table 31
314 119870 cong 119878119906119911
3141119883 = 2119860 119905 = ((119886119887)21198861198872)6 See Table 32
Table 31
119883119862 |O| 119892
2119860 455 119887119886119887311988611988721198861198873119886119887
2119861
455 11988711988611988731198861198872119886119887119886119887119886119887119886119887119886119887
455 11988711988611988711988611988711988611988711988611988711988611988721198861198873119886119887
1 1198873119886119887311988611988731198861198872119886119887119886119887119886119887
211988611988721198861198873119886119887119886119887211988611988721198861198872
1 1198871198861198872119886119887119886119887311988611988711988611988721198861198872119886119887119886119887211988611988711988611988731198861198873119886119887119886119887119886119887
31198861198871198861198872119886119887119886119887211988611988731198861198871198861198873
3119860 5824 1198872119886119887119886119887119886119887119886119887119886119887
4119860 3640 (119886119887)10119886(119886119887)15119886((119886119887)
5119886)2
4119862 3640 1198873119886119887119886119887119886119887
21198861198872
4119863 3640 119887311988611988711988611988721198861198873119886119887
51198605824 119887
311988611988711988611988721198861198871198861198872
5824 119887211988611988731198861198872119886119887
5119861 29120 1198872119886119887119886119887
6119860 29120 1198872119886119887119886119887119886119887
2119886119887
7119860
29120 11988721198861198872
29120 119887
29120 1198872
1011986029120 119887119886119887119886119887
2119886119887119886119887
29120 119887119886119887119886119887
1011986129120 119887119886119887119886119887
31198861198873119886119887
29120 11988711988611988731198861198873119886119887119886119887
12119860 58240 1198873119886119887119886119887119886119887
2119886119887
13119860
29120 119887119886119887
29120 119887119886119887119886119887119886119887
29120 11988731198861198872119886119887
1411986029120 119887
311988611988721198861198872119886119887119886119887
29120 11988711988611988721198861198872119886119887119886119887119886119887
1411986129120 119887
21198861198872119886119887119886119887119886119887
29120 11988711988611988711988611988711988611988721198861198872
1411986229120 119887119886119887
2119886119887119886119887119886119887119886119887
2
29120 1198872119886119887119886119887119886119887119886119887
2119886119887
15119860 116480 11988731198861198872119886119887119886119887119886119887
2011986058240 119887
31198861198871198861198872
58240 11988721198861198873119886119887
2611986029120 119887119886119887
2
29120 1198872119886119887
2611986129120 119887119886119887
21198861198872
29120 11988721198861198872119886119887
2611986229120 119887
3119886119887119886119887119886119887
29120 11988711988611988711988611988721198861198872
29119860 116480 1198871198861198872119886119887
29119861 116480 1198873119886119887119886119887
Table 32
119883119862
|O| 119892
2119860360 119886
54 ((119886119887)2119886)4119886((119886119887)
2119886)3
3119861 5120 11988721198861198872119886119887119886119887119886119887119886119887119886119887119886
3119862 9216 119887119886119887119886
4119860 1728 119887211988611988721198861198872119886119887119886119887119886119887
4119862 17280 1198872119886119887
5119861 55296 1198872119886119887119886119887119886
6119863 46080 119887
10 Algebra
Table 33119883119862|O| 119892
2119860 315 1198871198861198872119886119887119886119887119886119887
21198861198872119886119887211988611988721198861198872119886119887119886119887119886119887
2119886119887
2119861
630 1198872119886119887119886119887119886119887119886119887
2119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
21198861198872119886119887119886119887211988611988721198861198872119886119887
560 119887119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
2 (1198871198861198872119886)2(1198872119886)2(119887119886(1198872119886)2)2(1198872119886119887119886)2((1198872119886)21198872119886119887119886)21198871198861198872(119886119887)2
3119860 224 (1198871198861198872119886)3119887(119887119886)3(1198872119886)2(119887119886)5119887(119887119886)4(1198872119886)2119887
3119861 2240 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887
3119862 20160 11988711988611988721198861198871198861198872119886119887
41198602520 119887119886119887119886119887119886119887
2119886119887211988611988721198861198872119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886119887119886119887
4119861
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
2520 11988711988611988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
21198861198872119886119887
1260 119887211988611988721198861198871198861198872119886119887119886119887
4119862
5040 11988711988611988721198861198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
2520 11988711988611988711988611988711988611988721198861198872119886119887119886119887119886119887
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
4119863
10080 11988711988611988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
10080 11988711988611988711988611988721198861198872119886119887119886119887211988611988721198861198872119886119887
1120 (119886119887)4((119886119887)4119886)4119886(119886119887)8((119886119887)4119886)2
1120 (119886(1198872119886)4)2(1198872119886)8119886(119886(1198872119886)4)4(1198872119886)4
5119860 20160 119887119886119887119886119887
5119861 40320 1198872119886119887119886119887119886119887
2119886119887
6119860 10080 1198871198861198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
6119863 20160 11988711988611988721198861198872119886119887
611986440320 119887
2119886119887119886119887119886119887
2119886119887119886119887119886119887
40320 1198872119886119887119886119887211988611988721198861198872119886119887119886119887119886119887
7119860 80640 1198872119886119887119886119887119886119887119886119887
811986020160 119887119886119887
2119886119887119886119887211988611988721198861198872119886119887
20160 1198871198861198872119886119887211988611988721198861198871198861198872119886119887
8119861
20160 1198871198861198872119886119887119886119887119886119887119886119887
211988611988721198861198871198861198872119886119887
20160 11988711988611988711988611988721198861198872119886119887
20160 11988711988611988721198861198872119886119887119886119887
20160 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887119886119887
2119886119887
10080 11988721198861198872119886119887
10080 1198872119886119887119886119887
8119862
20160 11988721198861198871198861198872119886119887119886119887119886119887
20160 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
2119886119887
20160 1198872119886119887211988611988721198861198871198861198872119886119887
20160 1198871198861198872119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
10119860
40320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
119887211988611988711988611988721198861198871198861198872119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886119887
20160 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887
20160 11988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
1011986140320 119887119886119887119886119887119886119887119886119887
2119886119887
40320 1198871198861198872119886119887119886119887119886119887119886119887
11119860 161280 11988721198861198871198861198872119886119887119886119887
1211986040320 119887
2119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
20160 119887211988611988711988611988721198861198872119886119887119886119887119886119887
2119886119887
1211986140320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886119887
11988721198861198872119886119887211988611988721198861198872119886119887
12119862 40320 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
1211986380640 119887119886119887
2119886119887119886119887119886119887
80640 1198871198861198871198861198871198861198872119886119887
13119860 161280 119887119886119887
13119861 161280 11988721198861198872119886119887119886119887
14119860 80640 119887211988611988711988611988721198861198872119886119887119886119887
15119860 161280 119887
15119861 161280 1198872119886119887
15119862 161280 119887211988611988721198861198871198861198872119886119887119886119887119886119887
Table 33 Continued119883119862
|O| 119892
2011986080640
80640
1003816100381610038161003816100381610038161003816100381610038161003816
119887119886119887211988611988721198861198872119886119887119886119887
119887119886119887119886119887211988611988721198861198872119886119887
21119860 161280 119887119886119887211988611988721198861198872119886119887119886119887119886119887
21119861 161280 119887119886119887211988611988721198861198872119886119887
2411986080640 119887119886119887119886119887
2119886119887
80640 1198871198861198872119886119887119886119887
Table 34
119883119862
|O| 119892
2119860 315 ((119886119887)3119886)2119886((119886119887)
3119886)4119886(119886119887)
3((119886119887)3119886)2119886
2119861 1800 1198861198871198861198871198861198872
3119860 200 119887211988611988721198861198871198861198872119886119887119886119887
3119862 16800 11988611988721198861198872119886119887119886119887119886
41198603150 119887119886119887
21198861198871198861198872119886
630 1198871198861198872119886119887119886119887119886119887119886119887119886119887
2119886
4119862 12600 1198871198861198872119886119887119886
5119861 20160 119886119887119886
6119860 12600 1198861198871198861198871198861198872119886119887
6119864 100800 119887119886
7119860 100800 119887
811986025200 119886119887119886119887119886119887
25200 119887119886119887119886119887119886119887
12119860 50400 119886
Table 35
119883119862
|O| 119892
2119860 495 (119886119887)5119886 (119886119887)
2(119886 (119886119887)
5)4
(119886119887)5
2119861 792 11988721198861198871198861198872119886119887119886119887119886119887
2119886119887119886119887
3119860 264 119886(119886119887)9((119886119887)9119886)4119886(119886119887)
9((119886119887)9119886)2
31198615280 119886(119886119887)
6((119886119887)3119886)4
1760 (119886((119886119887)3119886)2)4119886
3119862 2640 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886
4119860
2970 11988711988611988711988611988721198861198871198861198872119886119887119886
495
495
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886
4119861 11880 119887119886119887211988611988721198861198871198861198872119886119887119886
4119862 11880 1198872119886119887
5119860 19008 119887119886119887119886
5119861 47520 1198861198872119886119887119886119887119886
6119860
11880 119886119887119886
3960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198871198861198872119886
11988611988711988611988721198861198872119886
611986115840 119886119887
2119886119887119886119887
15840 1198861198872119886119887119886119887119886119887119886119887119886119887
611986215840 119887
21198861198872119886119887119886
15840 1198872119886119887119886119887119886119887119886119887119886119887
21198861198872
611986347520 119886119887
2119886119887119886119887119886119887119886
15840 1198861198872119886119887119886119887119886119887
2119886119887119886
Algebra 11
Table 35 Continued
119883119862
|O| 119892
6119864 15840 119886119887119886119887119886119887119886119887211988611988711988611988721198861198872119886119887
7119860 47520 1198871198861198872119886119887119886
811986147520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886
119886119887119886119887119886119887
911986095040 119887119886119887119886119887119886119887
31680 119887119886119887119886119887211988611988721198861198871198861198872119886
911986195040 119887
2119886119887119886119887119886
31680 1198861198872119886119887119886119887119886119887119886119887119886119887119886
10119860 95040 119886119887
1011986147520 119887119886119887119886119887
21198861198872119886
47520 11988711988611988721198861198872119886119887119886
11119860 190080 119886
12119860 47520 1198872119886119887119886
12119861
47520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198872119886119887119886119887119886119887
2
119886119887119886119887119886119887119886119887119886
15840
15840
1003816100381610038161003816100381610038161003816100381610038161003816
11988721198861198872119886119887119886119887
119887119886119887119886119887119886
1211986247520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198872119886119887119886
1198861198871198861198871198861198871198861198871198861198871198861198872
12119864
47520 11988721198861198872119886
47520 119886119887119886119887
47520 119887119886119887119886119887119886119887119886119887
47520 1198872119886119887119886119887119886119887119886
1411986047520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198871198861198872119886119887119886119887
119887
15119862 190080 119886119887119886119887119886
1811986095040 119887119886119887
2119886119887119886119887119886
95040 11988611988721198861198872119886119887119886119887
1811986195040 119887
21198861198872119886119887119886119887119886
95040 1198871198861198871198861198871198861198872119886
2011986095040
95040
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198872119886119887119886
119887119886
Table 36
119883119862
|O| 119892
2119860 630 119886119887211988621198872119886
3119861 8960 11988611988721198862
3119862 1920 11988621198871198862
Table 37
119883119862
|O| 119892
4119861 30240 1198861198872119886
5119861 48384 1198861198871198862
6119862 80640 119886
Table 38
119883119862
|O| 119892
2119860 495 119887119886119887119886211988711988621198871198861198872119886119887
2119861396 119886119887
311988611988721198862119887119886
396 11988621198873119886119887211988621198871198862
31198601320 119887119886
211988711988621198872119886119887
440 11988611988721198861198871198861198872119886119887119886119887
31198612640 119886
211988721198861198871198861198873119886
1584 119886119887119886119887119886211988731198862
3119862 7920 11988711988611988721198861198872119886119887
4119860 2970 1198871198861198871198861198871198861198872119886119887119886
4119861
11880 1198862119887
5940 1198872
2970 1198871198861198871198861198872119886119887
51198609504 119887
21198861198871198862
9504 119887119886211988731198862119887119886
5119861 47520 11988621198872119886
611986011880 119887119886
211988721198861198872119886
3960 119886119887211988621198871198861198871198862119887119886
611986115840 119886
2
15840 119886
6119862
23760 1198871198861198871198861198873119886
23760 11988711988621198872119886119887119886
15840 1198871198861198873119886119887119886
611986423760 119887
2119886
23760 11988621198872
711986095040 119887119886
2
31680 1198871198861198872119886119887119886
811986023760 119887119886119887119886119887
2119886
23760 1198872119886119887119886119887119886119887
811986123760 119886119887119886119887
2119886
23760 11988721198862119887119886
3142 119883 = 2119861 119905 = 119886 See Table 33
3143 119883 = 2119862 119905 = 119888 See Table 34
3144 119883 = 2119863 119905 = (119888(119888119889)12)11 See Table 35
315 119870 cong 1198621199003
3151 119883 = 2119860 119905 = 1198872 See Tables 36 and 37
12 Algebra
Table 39
119883119862
|O| 119892
8119862
23760 11988611988721198861198872
23760 1198872119886211988721198862
23760 1198872119886119887119886
23760 11988711988621198871198861198873
911986031680 119887119886119887
21198861198873
31680 119886119887119886119887119886119887119886119887
9119861 63360 1198861198871198861198871198861198872
1011986047520 119886119887119886
21198872
47520 1198871198861198871198861198872
1011986147520 119886119887119886
47520 119886211988731198862
1211986023760 119886
21198871198862
23760 1198861198873119886
12119861
47520 119887
47520 1198861198871198861198871198862
47520 1198871198861198872119886119887
1411986095040 119887119886119887119886
95040 1198861198872119886119887119886
1511986095040 119886119887
2
95040 119886119887119886119887
15119861 190080 1198861198871198861198872
1811986095040 119886119887
2119886119887
95040 119887119886119887119886119887
21119860 190080 11988711988621198871198862
2411986095040 119886119887
2119886
95040 119886211988721198862
2411986195040 119886
2119887119886
95040 119887119886
3011986095040 119886119887119886
2
95040 11988611988731198862
3152 119883 = 2119861 119905 = ((1198862119887)3119886119887)5 See Tables 38 and 39
316 119870 cong 1198621199002
3161 119883 = 2119860 119905 = 119886 See Table 40
3162 119883 = 2119861 119905 = 119886(119886119887119886119887119886)2119886(119886119887119886119887119886)7 See Table 41
3163 119883 = 2119862 119905 = (119887minus1(1198861198872)2119886119887minus1(1198861198872)4)5 See Table 42
Table 40
119883119862
|O| 119892
2119861 1260 1198871198861198872119886119887
2119862 1008 11988731198861198871198861198873
3119861 14336 119887119886119887
4119862 40320 119887
Table 41
119883119862
|O| 119892
2119860 15 (1198874119886)2119887311988611988721198861198874(119887119886119887)
2119886(119887119886119887)
711988611988711988611988731198861198872(119886119887)2
21198611680 119887
31198861198871198861198873
210 11988741198861198874119886119887119886119887
2119862 2520 (119886119887)5119886(119886119887)
2119886(119886119887)
3
3119861 35840 1198872119886119887
4119860 1920 11988711988611988731198861198872119886119887119886119887
4119862 20160 1198871198861198872119886119887
4119864 161280 119887119886119887
411986513440 119887119886119887119886119887119886119887119886119887
13440 119887311988611988711988611988721198861198872
5119861 344064 1198872
6119864 430080 119887
Table 42
119883119862
|O| 119892
2119860240 119886119887
4119886119887211988611988711988611988741198861198873119886119887311988611988741198861198873119886119887119886
45 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887411988611988741198861198872119886
2119861
720 1198872119886
240 11988741198861198874119886119887311988611988741198861198874119886119887211988611988731198861198872119886
180 119886119887211988611988731198861198872119886119887119886119887311988611988731198861198872
30 1198861198871198861198874119886
2119862
1440 1198861198871198861198873119886119887311988611988721198861198871198861198872119886
1440 119886119887311988611988741198861198873119886119887211988611988721198861198874119886
1440 1198872119886119887119886119887119886119887
311988611988741198861198873
1152 11988611988711988611988731198861198873119886119887311988611988741198861198872119886
360 11988611988711988611988711988611988731198861198872119886119887119886119887311988611988731198861198872
3119860 10240 1198861198874119886119887211988611988711988611988721198861198871198861198872119886
3119861
15360 11988611988731198861198872119886
7680 11988731198861198872
1024 1198861198872119886119887211988611988731198861198873119886
41198603840 119887
411988611988741198861198873119886119887411988611988741198861198874
1440 1198861198871198861198872119886119887211988611988741198861198872
4119861
11520 1198871198861198874119886119887411988611988731198861198873119886
5760 119886119887211988611988711988611988721198861198871198861198873119886
5760 11988611988721198861198874119886119887311988611988741198861198873119886
1440 11988611988741198861198871198861198872119886119887411988611988721198861198871198861198873119886
1440 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887119886
720 119886119887119886119887411988611988721198861198872119886119887211988611988731198861198873
720 1198872119886119887411988611988721198861198872119886119887411988611988731198861198874119886
Algebra 13
Table 42 Continued
119883119862
|O| 119892
4119862
11520 11988611988721198861198873119886
11520 1198861198873119886119887119886119887311988611988721198861198872
5760 119887311988611988741198861198872119886
5760 11988611988731198861198871198861198872
2880 11988611988711988611988741198861198872119886
1440 119887411988611988741198861198873119886119887211988611988721198861198873119886
1440 1198861198872119886119887311988611988731198861198872119886119887119886119887
720 11988611988711988611988711988611988711988611988721198861198871198861198873119886
720 119886119887311988611988741198861198872119886
4119863
23040 1198871198861198873119886119887119886119887211988611988731198861198872
11520 119886119887119886119887311988611988731198861198871198861198873
5760 119886119887411988611988711988611988741198861198872119886
2880 11988611988711988611988711988611988741198861198872119886119887411988611988741198861198873119886
2880 1198861198872119886119887119886119887119886119887
311988611988711988611988741198861198874119886
4119864
23040 1198871198861198871198861198871198861198872
23040 119886119887311988611988721198861198872
11520 1198861198873119886119887211988611988741198861198872
11520 119887311988611988711988611988731198861198872119886
5760 11988731198861198874119886
5760 1198861198871198861198872
5760 1198874119886119887311988611988721198861198871198861198873119886
4119865
23040 119886119887119886119887211988611988721198861198872
23040 1198873119886119887311988611988731198861198874119886
11520 119887411988611988741198861198872119886119887411988611988731198861198872119886
11520 119886119887311988611988741198861198873119886119887211988611988721198861198872
5760 1198872119886119887411988611988721198861198872119886119887211988611988731198861198872119886
5760 1198872119886119887211988611988731198861198873119886119887411988611988721198861198873119886
5760 1198861198873119886119887211988611988741198861198872119886119887311988611988731198861198872119886
5760 1198861198874119886119887119886119887119886119887
3119886119887211988611988721198861198873
5760 119886119887211988611988731198861198871198861198872119886119887211988611988731198861198873
5760 1198874119886119887411988611988721198861198872119886119887211988611988731198861198872119886
2880 119887311988611988721198861198871198861198872
2880 1198873119886119887411988611988731198861198872
1920 11988611988711988611988741198861198872119886119887211988611988741198861198873
1920 1198871198861198873119886119887311988611988711988611988721198861198874
4119866
46080 1198873119886119887311988611988721198861198873119886
46080 1198861198872119886119887311988611988721198861198872
23040 1198861198871198861198872119886119887211988611988731198861198872119886
23040 11988611988731198861198872119886119887311988611988731198861198874119886
5119860 73728 119886119887211988611988731198861198871198861198872119886
5119861184320 119887
21198861198872119886
122880 119887211988611988731198861198872
Table 42 Continued
119883119862
|O| 119892
6119860 92160 1198871198861198873119886119887411988611988731198861198872119886
6119861 122880 119887211988611988741198861198872119886
6119862
46080 1198872119886119887311988611988731198861198872119886
46080 119887311988611988711988611988721198861198872
15360 1198874119886119887211988611988731198861198873119886
15360 119886119887211988611988721198861198873119886119887
6119863
92160 119887411988611988711988611988721198861198874119886
23040 11988741198861198873119886119887411988611988721198861198872
15360 11988611988721198861198872119886119887119886
15360 119886119887411988611988731198861198873119886
6119864
92160 1198871198861198871198861198872
92160 1198871198861198874119886
46080 1198873119886119887211988611988731198861198872119886
46080 11988711988611988721198861198873119886
46080 11988721198861198871198861198874119886
46080 11988611988711988611988741198861198873
15360 119886119887211988611988711988611988741198861198871198861198872119886
6119865
92160 1198861198873119886119887119886
92160 11988611988741198861198872119886
46080 119887311988611988721198861198871198861198872119886
46080 1198861198874119886119887411988611988731198861198873
46080 1198861198873119886119887411988611988731198861198872
46080 11988721198861198872119886119887119886119887119886
7119860 737280 11988741198861198872
811986092160 119887
2119886119887119886
92160 11988611988741198861198873
8119861
92160 119886119887411988611988721198861198872119886119887119886
92160 119886119887411988611988731198861198873119886119887119886
46080 11988611988711988611988731198861198872119886
46080 119886119887311988611988721198861198874119886
23040 119887119886119887311988611988731198861198871198861198873
23040 11988721198861198874119886119887211988611988721198861198874
8119862
92160 1198873119886119887311988611988731198861198874
92160 1198872
92160 1198873
92160 119887119886119887211988611988721198861198872
8119863
92160 119887311988611988731198861198872
92160 11988611988721198861198873119886119887311988611988741198861198872
92160 119887411988611988731198861198873119886
92160 11988711988611988721198861198871198861198873119886
92160 119887311988611988721198861198872
14 Algebra
Table 42 Continued
119883119862
|O| 119892
8119864
92160 1198861198873119886
92160 1198861198872119886
46080 119886119887311988611988711988611988741198861198872
46080 119887311988611988711988611988741198861198872119886
46080 119886119887211988611988721198861198871198861198873
46080 1198872119886119887119886119887119886
46080 1198872119886119887411988611988731198861198873119886
46080 1198872119886119887211988611988731198861198873119886
46080 1198861198872119886119887211988611988731198861198873
46080 119886119887411988611988741198861198873
23040 1198861198874119886119887211988611988721198861198871198861198873
23040 11988741198861198874119886119887211988611988741198861198873
8119865
184320 11988611988741198861198871198861198873119886
184320 11988611988721198861198874119886119887119886
92160 119886119887119886119887311988611988711988611988721198861198872
92160 11988731198861198874119886119887411988611988731198861198872119886
9119860737280 119886119887
41198861198871198861198872
245760 119887211988611988741198861198872
10119860 368640 119887211988611988731198861198872119886
10119861368640 119886119887
21198861198872
368640 11988731198861198873119886
10119862368640 119886119887
4
368640 11988611988711988611988731198861198871198861198873119886
11119860 1474560 11988721198861198873119886
12119860122880 119887119886119887
21198861198871198861198874119886
122880 119887411988611988711988611988741198861198873
12119861368640 119887
211988611988711988611988731198861198872
184320 1198861198874119886119887211988611988741198861198872
12119862368640 119886119887
2119886119887211988611988731198861198872
184320 119886119887411988611988711988611988721198861198872
12119863
184320 119887411988611988711988611988721198861198872
184320 11988611988711988611988711988611988731198861198873119886
92160 119886119887411988611988721198861198873119886
92160 11988611988721198861198873119886119887119886
12119864368640 119887
311988611988721198861198874
368640 11988711988611988731198861198872
12119865
184320 11988721198861198871198861198872119886
184320 11988611988731198861198874119886
184320 11988611988711988611988731198861198873
184320 1198861198871198861198872119886
92160 1198861198872119886119887119886
92160 11988611988741198861198873119886
12119866 368640 11988711988611988741198861198872119886
Table 42 Continued
119883119862
|O| 119892
12119867
184320 1198874119886119887311988611988731198861198872
184320 119886119887311988611988721198861198873119886
184320 11988611988711988611988711988611988741198861198872119886
184320 1198861198874119886119887311988611988741198861198872119886
184320 119887311988611988721198861198872119886119887
184320 119886119887211988611988731198861198872119886
14119860 737280 1198861198874119886
15119860737280 119887
4119886
737280 1198871198861198872119886
16119860
368640 11988611988731198861198872
368640 11988611988721198861198871198861198874
368640 11988731198861198872119886
368640 11988711988611988741198861198873119886
18119860 737280 11988731198861198873
20119860737280 119886119887
41198861198872
737280 1198873119886119887119886
20119861737280 119886119887
41198861198874119886
737280 119886119887119886119887119886
24119860737280 119886119887
2
737280 1198873119886
24119861737280 119887
4
737280 119887
28119860737280 119886119887119886119887
31198861198872
737280 119886119887411988611988721198861198872
30119860737280 119887
21198861198874
737280 1198871198861198873
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R H Bing ldquoA homeomorphism between the 3-sphere andthe sum of two solid horned spheresrdquo Annals of MathematicsSecond Series vol 56 pp 354ndash362 1952
[2] D Montgomery and L Zippin ldquoExamples of transformationgroupsrdquo Proceedings of the American Mathematical Society vol5 pp 460ndash465 1954
[3] J W Morgan and H Bass Eds The Smith Conjecture vol 112of Pure and Applied Mathematics Academic Press OrlandoFla USA 1984 Papers presented at the symposium held atColumbia University New York NY USA 1979
[4] C M Edwards and G T Ruttimann ldquoInvolutive and Peircegradings in JBWlowast-triplesrdquo Communications in Algebra vol 31no 6 pp 2819ndash2848 2003
Algebra 15
[5] J Gornicki and B E Rhoades ldquoA general fixed point theoremfor involutionsrdquo Indian Journal of Pure and Applied Mathemat-ics vol 27 no 1 pp 13ndash23 1996
[6] S Montgomery Fixed Rings of Finite Automorphism Groupsof Associative Rings vol 818 of Lecture Notes in MathematicsSpringer Berlin Germany 1980
[7] T A Springer ldquoThe classification of involutions of simplealgebraic groupsrdquo Journal of the Faculty of Science University ofTokyo Section IA Mathematics vol 34 no 3 pp 655ndash670 1987
[8] T A Springer ldquoSome results on algebraic group s with invo-lutionsrdquo in Algebraic Groups and Related Topics (KyotoNagoya1983) vol 6 of Advanced Studies in Pure Mathematics pp 525ndash543 North-Holland Amsterdam The Netherlands 1985
[9] W Feit and J GThompson ldquoSolvability of groups of odd orderrdquoPacific Journal of Mathematics vol 13 pp 775ndash1029 1963
[10] R Brauer and K A Fowler ldquoOn groups of even orderrdquo Annalsof Mathematics vol 62 no 3 pp 565ndash583 1955
[11] D Gorenstein The Classification of Finite Ssimple Groups Vol1 Groups of Noncharacteristic 2 Type The University Series inMathematics Plenum Press New York NY USA 1983
[12] D Gorenstein R Lyons and R Solomon The Classification ofthe Finite Simple Groups vol 40 of Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 1994
[13] C Bates and P Rowley ldquoInvolutions in Conwayrsquos largest simplegrouprdquo LMS Journal of Computation andMathematics vol 7 pp337ndash351 2004
[14] P Rowley and P Taylor ldquoPoint-line collinearity graphs of twosporadic minimal parabolic geometriesrdquo Journal of Algebra vol331 pp 304ndash310 2011
[15] P Rowley and P Taylor ldquoInvolutions in Jankorsquos simple group 1198694rdquoLMS Journal of Computation and Mathematics vol 14 pp 238ndash253 2011
[16] P Taylor ldquoInvolutions in Fischerrsquos sporadic groupsrdquo preprinthttpeprintsmamanacuk1622
[17] The GAP Group GAP-Groups Algorithms Programming Ver-sion 43 2002 httpwwwgap-systemorg
[18] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I The user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997
[19] R Wilson P Walsh J Tripp et al ldquoAtlas o f finite grouprepresentationsrdquo httpbrauermathsqmulacukAtlasv3
[20] J H Conway R T Curtis S P Norton R A Parker and R AWilson Atlas of Finite Groups Clarendon Oxford UK 1985
[21] R AWilson ldquoStandard generators for sporadic simple groupsrdquoJournal of Algebra vol 184 no 2 pp 505ndash515 1996
[22] D Gorenstein Finite Groups Chelsea Publishing New YorkNY USA 2nd edition 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Algebra 3
Table 1
119866 2119860 2119861 2119862 2119863
11987211
811987212
10 11 131198691
2211987222
13 5 101198692
6 31 1411987223
9119867119878 9 27 5 321198693
27 1711987224
14 26119872119888119871 6 28119867119890 12 34 41119877119906 22 42119878119906119911 9 66 15 411198621199003
7 611198621199002
5 13 178
Table 2
119883119862
|O| 119892
2119860 12 1198871198861198873119886119887
311986024 119887
2
8 1198873119886119887
4119860 24 119887119886119887
5119860 48 119887119886119887119886119887
611986024 119887
24 1198873
Table 3
119883119862
|O| 119892
211986010
10
1003816100381610038161003816100381610038161003816100381610038161003816
11988721198861198871198861198872119886119887119886119887
119887211988611988721198861198871198861198872119886119887
2119861 15 119886
3119861 60 119886119887
4119860 30 1198861198872119886119887119886
4119861 30 1198871198861198871198861198871198861198872119886
5119860 120 119887119886
611986060
60
1003816100381610038161003816100381610038161003816100381610038161003816
119887
1198872
two 119862119870(119905)-orbits in 119883
119862signifies that their union is a 119862
119866(119905)-
orbit
31 Permutation Ranks See Table 1
32 119870 cong 11987211
321119883 = 2119860 119905 = 119886 See Table 2
33 119870 cong 11987212
331 119883 = 2119860 119905 = ((119886119887)21198861198872)3 See Table 3
Table 4
119883119862
|O| 119892
211986124 119887119886119887
2119886119887
6 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886119887119886119887
311986032
32
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887
1198871198861198872
3119861 16 11988711988611988721198861198872119886119887119886119887
4119860 48 1198871198861198871198861198872
4119861 48 1198872119886119887119886119887
5119860 96 119887119886119887
611986196
96
1003816100381610038161003816100381610038161003816100381610038161003816
119887
1198872
Table 5
119883119862
|O| 119892
211986015 119887119886119887
2119886119887119886119887119886119887
1 119887119886119887119886119887119886119887211988611988721198861198871198861198872119886119887
2119861 15 1198871198861198871198861198872
311986020
20
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887
1198861198871198861198872119886119887119886
3119861 60 1198871198861198872
511986060 119886119887119886119887
2119886
60 119886119887
6119860 60 119887119886119887119886
611986160
60
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887
119886
1011986060 119887119886119887
2119886119887119886
60 119887119886119887119886119887
11119860 120 119887
11119861 120 119887119886
332 119883 = 2119861 119905 = 119886 See Table 4
333 119883 = 2119862 119905 = 119888 See Table 5
34 119870 cong 1198691
341 119883 = 2119860 119905 = 119886 See Table 6
35 119870 cong 11987222
351 119883 = 2119860 119905 = 119886 See Table 7
352 119883 = 2119861 119905 = 119888 See Table 8
353 119883 = 2119862 119905 = (1198881198891198881198892)5 See Table 9
36 119870 cong 1198692
361 119883 = 2119860 119905 = (1198871198861198872(119886119887)2119887(119886119887)3)5 See Table 10
4 Algebra
Table 6
119883119862
|O| 119892
211986015 119887119886119887
2119886119887119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
15 119887119886119887211988611988721198861198871198861198872119886119887119886119887119886119887
2119886119887119886119887
311986060 119887119886119887
2119886119887119886119887119886119887
2119886119887
12 1198872119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
511986060 119887119886119887119886119887
21198861198872119886119887119886119887
20 1198872119886119887119886119887119886119887
21198861198872119886119887
511986160 119887119886119887119886119887
21198861198871198861198872119886119887119886119887
20 11988721198861198871198861198872119886119887119886119887
611986060 119887119886119887
2119886119887119886119887
60 1198871198861198871198861198872119886119887
7119860 120 11988711988611988721198861198872119886119887119886119887
1011986060 119887
2119886119887119886119887119886119887
60 1198871198861198871198861198871198861198872
1011986160 119887
2119886119887119886119887
60 1198871198861198871198861198872
11119860 120 1198871198861198871198861198872119886119887119886119887
15119860 120 11988721198861198872119886119887119886119887
15119861 120 119887119886119887
19119860 120 119887
19119861 120 1198872119886119887
19119862 120 11988721198861198871198861198872119886119887
Table 7
119883119862
|O| 119892
2119860
24 1198871198861198873119886119887
12 (119887119886)21198872119886119887119886119887
8 11988721198861198871198861198872(119886119887)3
6 1198873(119886119887)2(1198861198873)2119886119887
3119860 192 119887119886119887
4119860
48 1198871198861198872119886119887
48 1198873119886119887119886119887
48 1198871198861198871198861198873
411986196 119887
2119886119887119886119887
96 1198871198861198871198861198872
5119860 384 1198872
6119860 192 119887
Table 8
119883119862
|O| 119892
211986042 119887
7 (1198872119886)2119887119886
3119860 112 119886
4119860 168 119886119887119886119887119886
Table 9
119883119862
|O| 119892
211986020 119887
2
5 11988711988611988721198861198873
3119860 80 119887
4119860 80 119887119886119887119886119887119886
411986140 119887119886119887
2119886
40 1198871198861198872119886119887
5119860 320 119886119887
611986080
80
1003816100381610038161003816100381610038161003816100381610038161003816
119886
1198861198872119886
11119860 320 1198872119886
11119861 320 119887119886
Table 10
119883119862
|O| 119892
2119860 10 119887119886
3119861 160 119887
4119860 80 119886
5119860 32 1198861198872119886119887119886
5119861 32 1198871198861198872
362 119883 = 2119861 119905 = 119886 See Table 11
363 119883 = 2119862 119905 = 119888 See Table 12
37 119870 cong 11987223
371119883 = 2119860 119905 = 119886 See Table 13
38 119870 cong 119867119878
381 119883 = 2119860 119905 = 119886 See Table 14
382 119883 = 2119861 119905 = ((119886119887)2119886)5 See Tables 15 and 16
383 119883 = 2119862 119905 = 119888 See Table 17
384 119883 = 2119863 119905 = ((119889119888)21198894119888119889)5 See Tables 18 and 19
39 119870 cong 1198693
391119883 = 2119860 119905 = 119886 See Table 20
392119883 = 2119861 119905 = 119888 See Table 21
310 119870 cong11987224
3101 119883 = 2119860 119905 = ((119886119887)31198861198872)4 See Tables 22 and 23
Algebra 5
Table 11
119883119862
|O| 119892
2119860 15 119887(119887119886)2119887(119887119886)
2119887(119887119886)
2
2119861
15
15
1003816100381610038161003816100381610038161003816100381610038161003816
((119887119886)2119887)3(119887119886)3
119887(119887119886)2(119887(119887119886)
31198872119886)2
1
1
1003816100381610038161003816100381610038161003816100381610038161003816
(119887119886)21198872119886(119887(119887119886)
2)21198872119886119887(119887119886)
3
1198872119886119887(119887119886)
3119887(119887119886)
2119887(119887119886)
3(1198872119886)2
3119860 24 119887(119887119886)2119887(119887119886)
2
3119861 60 119887(119887119886)3119887(119887119886)
2119887(119887119886)
3119887(119887119886)
4119860 60 (119887119886)119887(119887119886)2
5119860 60 (119887119886)2119887(119887119886)
2119887(119887119886)
3119887(119887119886)119887(119887119886)
2
5119861 60 (119887119886)119887(119887119886)119887(119887119886)3119887(119887119886)
2119887(119887119886)
3
5119862120 119887(119887119886)
3119887(119887119886)
3
24 119887(119887119886)119887(119887119886)2119887(119887119886)
3
5119863120 (119887119886)119887(119887119886)
2119887(119887119886)
3
24 (119887119886)2119887(119887119886)
2119887(119887119886)119887(119887119886)
6119860 120 (119887119886)3
611986160
60
1003816100381610038161003816100381610038161003816100381610038161003816
(119887119886)2119887(119887119886)
3
(119887119886)3119887(119887119886)
4
7119860 240 119887(119887119886)4
1011986060 119887(119887119886)119887(119887119886)
2119887(119887119886)
2
60 119887(119887119886)2119887(119887119886)
3
1011986160 (119887119886)
2119887(119887119886)
2119887(119887119886)
60 (119887119886)119887(119887119886)2119887(119887119886)119887(119887119886)
10119862120 (119887119886)119887(119887119886)119887(119887119886)
120 (119887119886)2119887(119887119886)
10119863120 119887(119887119886)
3
120 119887(119887119886)119887(119887119886)2
12119860120
120
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119887119886)
(119887119886)
15119860 240 (119887119886)119887(119887119886)119887(119887119886)2119887(119887119886)
2
15119861 240 (119887119886)119887(119887119886)2119887(119887119886)119887(119887119886)
2
Table 12
119883119862
|O| 119892
2119860 21 (1198871198861198872119886)2(119887119886)3
2119861 28 1198871198861198872
311986014
14
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)3(1198861198872)2119886119887
(1198872119886119887119886)2119887119886119887
3119861 168 119887119886119887119886
411986084 119886119887119886119887
2119886119887
42 1198872119886119887119886119887
6119860
84
84
1003816100381610038161003816100381610038161003816100381610038161003816
1198872
119887
42
42
1003816100381610038161003816100381610038161003816100381610038161003816
119886119887119886119887119886119887119886
1198861198871198861198872119886119887119886119887
6119861 168 119886119887
7119860 336 119887119886119887
8119860168 119886119887
2119886
168 119886119887119886
12119860168
168
1003816100381610038161003816100381610038161003816100381610038161003816
119887119886
119886
Table 13
119883119862
|O| 119892
211986084 119887119886119887
3119886119887
14 11988731198861198872119886119887
3119860 448 1198872
4119860
336 119887119886119887119886119887
336 119887119886119887119886119887119886119887
336 1198871198861198871198861198871198861198872
5119860 896 119887119886119887
6119860 1344 119887
Table 14
119883119862
|O| 119892
211986080 119887
211988611988731198861198872
30 119887211988611988711988611988721198861198871198861198872
3119860 640 119887119886119887
4119861 480 1198871198861198873119886119887
4119862 960 1198872119886119887
5119861 128 1198871198861198872119886119887
5119862 1536 119887
6119861 1920 1198872
Table 15
119883119862
|O| 119892
211986045 119887119886119887
3119886119887
30 11988741198861198874119886119887119886119887
211986136
36
1003816100381610038161003816100381610038161003816100381610038161003816
119887211988611988731198861198872
1198871198861198871198861198871198861198873119886
3119860360 119887
31198861198872119886119887119886119887
24 1198861198872119886119887211988611988711988611988721198861198872119886
4119861 180 119886
4119862
360 11988711988611988711988611988741198861198872119886
180
180
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198871198861198872
1198872119886119887119886119887119886119887
5119860144
144
1003816100381610038161003816100381610038161003816100381610038161003816
11988711988611988741198861198873119886119887
11988711988611988731198861198874119886119887
5119861 720 1198871198861198871198861198872119886
Table 16
119883119862
|O| 119892
6119861
720 119887119886119887
360 1198871198861198872119886
360 1198861198872119886119887
7119860 2880 119887
8119861720 119887
41198861198872119886
720 1198861198873119886119887
8119862720 119887119886119887
3119886
720 11988611988721198861198874
10119860720
720
1003816100381610038161003816100381610038161003816100381610038161003816
1198872
1198873
10119861720
720
1003816100381610038161003816100381610038161003816100381610038161003816
119886119887119886
119887119886
15119860 2880 1198872119886
6 Algebra
Table 17
119883119862
|O| 119892
2119860 105 119886
2119861 28 119887119886
3119860 336 1198872
4119861 630 119887
Table 18
119883119862
|O| 119892
211986060 119886119887
31198861198871198861198872119886
5 11988611988731198861198872119886
2119861 60 1198872119886
3119860240 119886
160 11988611988731198861198871198861198872119886119887119886
4119860 120 1198871198861198873119886119887119886
4119861
240 119886119887
120 1198871198861198872119886119887119886119887
30 1198861198872119886
4119862120 119886119887
21198861198872119886
120 1198861198873119886119887119886
5119860192 119886119887119886119887
3
192 11988711988611988721198861198873
5119861 320 1198874119886119887
5119862 1920 11988721198861198872
6119860480 119886119887
2119886119887
480 1198871198861198872119886
Table 19
119883119862
|O| 119892
6119861480 119887
4119886119887119886119887
480 11988721198861198872119886
7119860 1920 119887
8119860480 119887119886119887119886119887
2
480 11988711988611988721198861198872119886119887
10119860960 119887
211988611988721198861198872119886
960 119887119886119887
10119861 960 1198861198872
11119860 1920 1198872119886119887119886
11119861 1920 119887119886119887119886119887
12119860960 119887119886
960 1198873
15119860 1920 1198871198861198873
20119860960 119887119886119887119886
960 1198872119886119887
20119861960 119887
21198861198873119886
960 1198874119886119887119886
Table 20
119883119862
|O| 119892
2119860120 119887(119887119886)
3119887(119887119886)119887(119887119886)
5
10 (119887119886)119887(119887119886)4119887(119887119886)
2
3119860 192 119887(119887119886)6
3119861 640 (119887119886)119887(119887119886)2
4119860480 119887(119887119886)
5119887(119887119886)
80 (119887119886)3119887(119887119886)
3119887(119887119886)
4
5119860960 (119887119886)
3119887(119887119886)
3119887(119887119886)
2
320 (119887119886)2119887(119887119886)
2
5119861960 (119887119886)119887(119887119886)
3119887(119887119886)
4
320 (119887119886)119887(119887119886)3
6119860 960 (119887119886)119887(119887119886)6
8119860960 (119887119886)
4119887(119887119886)
2
960 (119887119886)119887(119887119886)5
9119860 1920 119887(119887119886)2119887(119887119886)
2
9119861 1920 (119887119886)
9119862 1920 119887(119887119886)2
10119860960 (119887119886)
4119887(119887119886)
960 119887(119887119886)4119887(119887119886)119887(119887119886)
10119861960 119887(119887119886)119887(119887119886)119887(119887119886)119887(119887119886)
2
960 119887(119887119886)5
12119860960
960
1003816100381610038161003816100381610038161003816100381610038161003816
(119887119886)2119887(119887119886)
4
(119887119886)6
15119860 1920 (119887119886)2119887(119887119886)
15119861 1920 119887(119887119886)3
17119860 1920 (119887119886)2
17119861 1920 (119887119886)2119887(119887119886)
3
Table 21
119883119862
|O| 119892
2119860 153 1198861198872119886119887119886119887119886119887
3119860102
102
1003816100381610038161003816100381610038161003816100381610038161003816
1198872
119887
3119861 272 1198861198871198861198872119886
4119860612 119887119886119887119886119887119886119887
2119886
306 1198871198861198872119886119887119886119887119886
6119860
612
612
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198872119886119887119886
1198871198861198871198861198872119886
306
306
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198872119886119887119886119887119886
119887119886119887119886
81198601224 119886119887119886119887119886119887
1224 1198872119886119887119886119887119886
9119860 816 11988721198861198871198861198872119886119887
9119861 816 1198871198861198871198861198872119886119887
9119862 816 1198871198861198872
121198601224
1224
1003816100381610038161003816100381610038161003816100381610038161003816
119886119887
1198861198872119886119887119886
17119860 2448 119887119886
17119861 2448 119886
19119860 2448 1198861198872119886119887
19119861 2448 1198872119886119887119886
Algebra 7
Table 22
119883119862
|O| 119892
2119860
168 1198872119886119887119886119887119886119887
2119886119887
14 119886119887119886
14 1198861198871198861198872119886119887
2119861 84 1198872119886119887
3119860 896 119887
3119861 128 119887211988611988721198861198871198861198872119886119887119886119887
Table 23
119883119862
|O| 119892
4119860112 119887119886119887119886119887119886119887
112 1198872119886119887211988611988721198861198872
4119861
1344 1198861198872119886119887119886
672 11988611988721198861198871198861198872119886119887
672 11988721198861198871198861198872119886119887119886
5119860 1792 119887119886119887119886
6119860 5376 119886
Table 24
119883119862
|O| 119892
211986060 119887
2119886119887119886119887211988611988721198861198871198861198872119886119887119886119887119886119887
2119886119887
15 11988711988611988721198861198872119886119887119886119887119886119887
2
2119861
120 119887119886119887211988611988711988611988721198861198871198861198872
80 119887211988611988711988611988721198861198871198861198872119886119887
2 (119887119886)8(1198872119886)21198871198861198872(119886119887)2(1198861198872)2(119886119887)3(1198861198872119886119887)2119887
3119860 320 119887211988611988721198861198871198861198872119886119887119886119887119886119887
3119861 960 11988721198861198871198861198872119886119887
4119860120 119887119886119887
21198861198872
120 1198871198861198871198861198872
4119861
480 1198872119886119887119886119887119886119887
2119886119887
240 1198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
240 1198872119886119887119886119887119886119887119886119887119886119887119886119887
4119862
960 1198871198861198872119886119887
960 11988721198861198871198861198872
160 1198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
21198861198872
160 1198872119886119887119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
5119860 1920 119887119886119887
6119860 1920 1198871198861198871198861198871198861198872
61198611920 119887119886119887
2
1920 1198872119886119887
101198601920 119887
21198861198871198861198872119886119887119886119887
1920 1198872119886119887119886119887119886119887119886119887
2
11119860 7680 119887119886119887119886119887
121198613840 119887
3840 1198872
3102 119883 = 2119861 119905 = 119886 See Table 24
311 119870 cong 119872119888119871
3111119883 = 2119860 119905 = 119886 See Table 25
3112119883 = 2119861 119905 = 119888 See Table 26
Table 25
119883119862
|O| 119892
2119860 210 119887211988611988731198861198872
3119861 2240 1198871198861198871198861198872
4119860 5040 1198872119886119887
5119861 8064 119887
6119861 6720 1198872
Table 26
119883119862
|O| 119892
2119860 165 11988611988721198861198871198861198872119886119887
3119860 220 1198872119886119887311988611988721198861198871198861198872119886
3119861660 119886119887
21198861198872
220 1198872119886119887119886119887119886119887
3
41198601980 119887
21198861198873119886119887119886119887119886
990 119886
5119860792 119887
211988611988721198861198873
792 1198872119886119887119886119887119886119887
2
5119861 7920 1198872
6119860 1980 119886119887
6119861
1980 1198871198861198872119886119887119886119887
1980 1198861198872119886119887
660
660
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198873
11988711988611988731198861198874119886119887
71198603960 119887119886119887119886119887
2119886
3960 1198871198861198872
71198613960 119887119886119887
3119886
3960 1198872119886
81198603960 119887
21198861198872119886
3960 11988611988731198861198873
9119860 2640 1198872119886119887119886
9119861 2640 1198861198871198861198872119886119887
101198603960 119887119886119887119886119887
3960 119886119887119886119887119886
121198603960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198873
1198871198861198873
141198603960 119886119887
21198861198873
3960 1198871198861198872119886119887
141198613960 119887119886119887119886119887119886
3960 1198872119886119887119886119887119886
151198603960 119887119886119887
3960 1198874
151198613960 119886119887119886119887
3960 119887
301198603960 119887119886
3960 1198871198861198872119886
301198613960 119887
41198861198872119886
3960 1198861198872
8 Algebra
Table 27
119883119862
|O| 119892
2119860240 119887119886119887119886119887119886119887
2 1198875(1198861198876)2(119886119887)31198875119886(11988721198861198873)2
2119861 315 119887211988611988711988611988721198861198872119886119887
3119860 1344 1198872119886119887
3119861 960 1198871198861198874119886119887
4119860480 119887
31198861198871198861198872119886119887
480 11988721198861198871198861198873119886119887
41198612520
2520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887
1198874119886119887
5119860 2688 1198873
6119860 13440 119887
Table 28
119883119862
|O| 119892
2119860 84 (119886119887)21198875(119886119887)2
2119861
168
168
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)2
(119886119887)1198874(119886119887)119887
2(119886119887)119887
2(119886119887)119887
4
14
14
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198874(119886119887)119887
4(119886119887)119887
2(119886119887)119887
2
1198872(119886119887)119887
2(119886119887)21198874(119886119887)119887
2(119886119887)119887
2
3119860 896 1198874(119886119887)119887
2
3119861 2688 1198872
4119861
1344
1344
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)119887
2(119886119887)119887
2
1198872(119886119887)119887
4(119886119887)21198874
112
112
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)1198872(119886119887)41198874(119886119887)2
(119886119887)31198874(119886119887)41198872
4119862
1344
1344
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)1198874(119886119887)119887
4
1198874(119886119887)119887
2(119886119887)119887
2(119886119887)31198872
672
672
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198872(119886119887)3
1198872(119886119887)41198872(119886119887)2
672
672
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)41198872(119886119887)2
(119886119887)4119887(119886119887)
2
5119860 5376 (119886119887)3
6119860 5376 (119886119887)1198872(119886119887)119887
2(119886119887)4
61198615376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198872(119886119887)
(119886119887)1198872(119886119887)
7119862 3072 1198872(119886119887)119887
2(119886119887)119887
2(119886119887)2
81198605376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)21198872(119886119887)21198872
(119886119887)1198872(119886119887)21198872(119886119887)
101198605376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)2
1198872(119886119887)119887
2(119886119887)119887
2
1211986110752
10752
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)2
1198874(119886119887)2
15119860 21504 1198872(119886119887)3
17119860 21504 (119886119887)1198872(119886119887)2
17119861 21504 (119886119887)
21119860 21504 (119886119887)1198872(119886119887)119887
2
21119861 21504 1198872(119886119887)
Table 29
119883119862
|O| 119892
2119860105 (119886119887)119887(119886119887)
3119887(119886119887)
3
63 119887611988611988731198861198873119886119887211988611988761198861198872119886119887611988611988761198861198874119886
2119861 315 (1198861198872)2(119887119886)21198872119886
3119860630 (119886119887)119887(119886119887)119887(119886119887)119887
126 119887(119886119887)21198876
3119861 2520 (119886119887)1198876
41198601260 (119886119887)
21198875(119886119887)
1260 (119886119887)119887(119886119887)1198873
4119861
945
945
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)21198872(119886119887)119887
119887(119886119887)119887(119886119887)1198872(119886119887)119887
315
315
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119886119887)5119887
119887(119886119887)31198872(119886119887)119887
41198621890
1890
1003816100381610038161003816100381610038161003816100381610038161003816
1198873(119886119887)2
(119886119887)1198872(119886119887)2119887
5119860 2520 (119886119887)3119887(119886119887)
6119860
3780 (119886119887)31198872(119886119887)2
1890 119887(119886119887)119887(119886119887)
1890 119887(119886119887)119887(119886119887)119887(119886119887)3
1260 119887(119886119887)1198872(119886119887)119887(119886119887)
2
6119861 7560 (119886119887)
71198601890 (119886119887)119887(119886119887)
41198872
90 119887(119886119887)1198873(119886119887)2119887
71198611890 119887
5(119886119887)119887(119886119887)
90 1198875(119886119887)119887
5(119886119887)119887
2(119886119887)119887
5
7119862 2160 119887(119886119887)119887(119886119887)4
71198637560 (119886119887)119887(119886119887)
4
1080 (119886119887)1198873(119886119887)
71198647560 (119886119887)
3119887(119886119887)
2
1080 (119886119887)1198874(119886119887)2119887(119886119887)
8119860
3780
3780
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)2119887(119886119887)
2
(119886119887)4
3780
3780
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)3119887(119886119887)119887(119886119887)119887
(119886119887)119887(119886119887)119887(119886119887)2
10119860 7560 (119886119887)3
121198607560 119887(119886119887)119887
7560 119887(119886119887)1198872
121198617560
7560
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119886119887)2119887
119887
14119860
3780 (119886119887)1198873(119886119887)2
3780 (119886119887)21198872(119886119887)
1890 (119886119887)1198872(119886119887)
1890 119887(119886119887)3119887
14119861
3780 (119886119887)119887(119886119887)2119887(119886119887)
3780 (119886119887)1198874
1890 (119886119887)21198872(119886119887)119887(119886119887)
2
1890 119887(119886119887)119887(119886119887)2
Algebra 9
Table 29 Continued
119883119862
|O| 119892
141198627560 119887(119886119887)
7560 119887(119886119887)3
141198637560 119887
3(119886119887)
7560 119887(119886119887)2
15119860 15120 (119886119887)1198872
21119860 15120 (119886119887)4119887(119886119887)
21119861 15120 (119886119887)119887
21119862 15120 (119886119887)3119887
21119863 15120 1198872
281198607560 119887
2(119886119887)119887
2
7560 (119886119887)2
281198617560 119887
4(119886119887)3
7560 (119886119887)119887(119886119887)2
Table 30
119883119862|O| 119892
2119860
640 11988611988731198861198872119886119887119886
480 11988611988721198861198872119886119887119886119887119886119887119886119887119886119887
21198861198872119886
240 11988611988721198861198873119886119887311988611988731198861198873119886119887211988611988711988611988721198861198873119886119887119886119887119886119887119886119887
31198861198873
30 1198861198872119886119887119886119887119886119887211988611988721198861198873119886119887311988611988711988611988711988611988721198861198873119886119887211988611988721198861198873119886119887211988611988711988611988711988611988731198863119860 10240 119886119887
3119886119887119886119887119886119887119886119887119886119887
2119886
4119860 5120 119886119887119886119887119886119887119886
4119862
7680 119886119887119886119887211988611988721198861198872119886119887119886
3840 119886119887211988611988731198861198871198861198872119886119887119886
3840 119886119887119886119887211988611988711988611988731198861198872119886
411986315360 119886119887119886
7680 1198861198872119886119887211988611988711988611988721198861198871198861198872119886
5119860 12288 11988611988731198861198871198861198872119886119887119886119887119886119887119886
5119861 4096 119886119887119886119887119886
611986061440 119886119887
21198861198871198861198872119886119887119886
30720 119886119887119886119887119886119887119886119887119886119887119886
7119860 61440 119886
811986261440 119886119887119886119887
21198861198872119886
61440 11988611988721198861198872119886119887119886
10119860 61440 11988611988721198861198872119886119887119886119887119886119887119886
12119860122880 1198861198872119886119887119886
13119860 61440 1198861198873119886119887119886
312 119870 cong 119867119890
3121119883 = 2119860 119905 = 119886 See Table 27
3122119883 = 2119861 119905 = (1198861198872)6 See Table 28
3123 119883 = 2119862 119905 = 1198893 See Table 29
313 119870 cong 119877119906
3131 119883 = 2119860 119905 = 1198872 See Table 30
3132 119883 = 2119861 119905 = 119886 See Table 31
314 119870 cong 119878119906119911
3141119883 = 2119860 119905 = ((119886119887)21198861198872)6 See Table 32
Table 31
119883119862 |O| 119892
2119860 455 119887119886119887311988611988721198861198873119886119887
2119861
455 11988711988611988731198861198872119886119887119886119887119886119887119886119887119886119887
455 11988711988611988711988611988711988611988711988611988711988611988721198861198873119886119887
1 1198873119886119887311988611988731198861198872119886119887119886119887119886119887
211988611988721198861198873119886119887119886119887211988611988721198861198872
1 1198871198861198872119886119887119886119887311988611988711988611988721198861198872119886119887119886119887211988611988711988611988731198861198873119886119887119886119887119886119887
31198861198871198861198872119886119887119886119887211988611988731198861198871198861198873
3119860 5824 1198872119886119887119886119887119886119887119886119887119886119887
4119860 3640 (119886119887)10119886(119886119887)15119886((119886119887)
5119886)2
4119862 3640 1198873119886119887119886119887119886119887
21198861198872
4119863 3640 119887311988611988711988611988721198861198873119886119887
51198605824 119887
311988611988711988611988721198861198871198861198872
5824 119887211988611988731198861198872119886119887
5119861 29120 1198872119886119887119886119887
6119860 29120 1198872119886119887119886119887119886119887
2119886119887
7119860
29120 11988721198861198872
29120 119887
29120 1198872
1011986029120 119887119886119887119886119887
2119886119887119886119887
29120 119887119886119887119886119887
1011986129120 119887119886119887119886119887
31198861198873119886119887
29120 11988711988611988731198861198873119886119887119886119887
12119860 58240 1198873119886119887119886119887119886119887
2119886119887
13119860
29120 119887119886119887
29120 119887119886119887119886119887119886119887
29120 11988731198861198872119886119887
1411986029120 119887
311988611988721198861198872119886119887119886119887
29120 11988711988611988721198861198872119886119887119886119887119886119887
1411986129120 119887
21198861198872119886119887119886119887119886119887
29120 11988711988611988711988611988711988611988721198861198872
1411986229120 119887119886119887
2119886119887119886119887119886119887119886119887
2
29120 1198872119886119887119886119887119886119887119886119887
2119886119887
15119860 116480 11988731198861198872119886119887119886119887119886119887
2011986058240 119887
31198861198871198861198872
58240 11988721198861198873119886119887
2611986029120 119887119886119887
2
29120 1198872119886119887
2611986129120 119887119886119887
21198861198872
29120 11988721198861198872119886119887
2611986229120 119887
3119886119887119886119887119886119887
29120 11988711988611988711988611988721198861198872
29119860 116480 1198871198861198872119886119887
29119861 116480 1198873119886119887119886119887
Table 32
119883119862
|O| 119892
2119860360 119886
54 ((119886119887)2119886)4119886((119886119887)
2119886)3
3119861 5120 11988721198861198872119886119887119886119887119886119887119886119887119886119887119886
3119862 9216 119887119886119887119886
4119860 1728 119887211988611988721198861198872119886119887119886119887119886119887
4119862 17280 1198872119886119887
5119861 55296 1198872119886119887119886119887119886
6119863 46080 119887
10 Algebra
Table 33119883119862|O| 119892
2119860 315 1198871198861198872119886119887119886119887119886119887
21198861198872119886119887211988611988721198861198872119886119887119886119887119886119887
2119886119887
2119861
630 1198872119886119887119886119887119886119887119886119887
2119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
21198861198872119886119887119886119887211988611988721198861198872119886119887
560 119887119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
2 (1198871198861198872119886)2(1198872119886)2(119887119886(1198872119886)2)2(1198872119886119887119886)2((1198872119886)21198872119886119887119886)21198871198861198872(119886119887)2
3119860 224 (1198871198861198872119886)3119887(119887119886)3(1198872119886)2(119887119886)5119887(119887119886)4(1198872119886)2119887
3119861 2240 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887
3119862 20160 11988711988611988721198861198871198861198872119886119887
41198602520 119887119886119887119886119887119886119887
2119886119887211988611988721198861198872119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886119887119886119887
4119861
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
2520 11988711988611988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
21198861198872119886119887
1260 119887211988611988721198861198871198861198872119886119887119886119887
4119862
5040 11988711988611988721198861198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
2520 11988711988611988711988611988711988611988721198861198872119886119887119886119887119886119887
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
4119863
10080 11988711988611988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
10080 11988711988611988711988611988721198861198872119886119887119886119887211988611988721198861198872119886119887
1120 (119886119887)4((119886119887)4119886)4119886(119886119887)8((119886119887)4119886)2
1120 (119886(1198872119886)4)2(1198872119886)8119886(119886(1198872119886)4)4(1198872119886)4
5119860 20160 119887119886119887119886119887
5119861 40320 1198872119886119887119886119887119886119887
2119886119887
6119860 10080 1198871198861198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
6119863 20160 11988711988611988721198861198872119886119887
611986440320 119887
2119886119887119886119887119886119887
2119886119887119886119887119886119887
40320 1198872119886119887119886119887211988611988721198861198872119886119887119886119887119886119887
7119860 80640 1198872119886119887119886119887119886119887119886119887
811986020160 119887119886119887
2119886119887119886119887211988611988721198861198872119886119887
20160 1198871198861198872119886119887211988611988721198861198871198861198872119886119887
8119861
20160 1198871198861198872119886119887119886119887119886119887119886119887
211988611988721198861198871198861198872119886119887
20160 11988711988611988711988611988721198861198872119886119887
20160 11988711988611988721198861198872119886119887119886119887
20160 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887119886119887
2119886119887
10080 11988721198861198872119886119887
10080 1198872119886119887119886119887
8119862
20160 11988721198861198871198861198872119886119887119886119887119886119887
20160 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
2119886119887
20160 1198872119886119887211988611988721198861198871198861198872119886119887
20160 1198871198861198872119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
10119860
40320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
119887211988611988711988611988721198861198871198861198872119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886119887
20160 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887
20160 11988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
1011986140320 119887119886119887119886119887119886119887119886119887
2119886119887
40320 1198871198861198872119886119887119886119887119886119887119886119887
11119860 161280 11988721198861198871198861198872119886119887119886119887
1211986040320 119887
2119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
20160 119887211988611988711988611988721198861198872119886119887119886119887119886119887
2119886119887
1211986140320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886119887
11988721198861198872119886119887211988611988721198861198872119886119887
12119862 40320 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
1211986380640 119887119886119887
2119886119887119886119887119886119887
80640 1198871198861198871198861198871198861198872119886119887
13119860 161280 119887119886119887
13119861 161280 11988721198861198872119886119887119886119887
14119860 80640 119887211988611988711988611988721198861198872119886119887119886119887
15119860 161280 119887
15119861 161280 1198872119886119887
15119862 161280 119887211988611988721198861198871198861198872119886119887119886119887119886119887
Table 33 Continued119883119862
|O| 119892
2011986080640
80640
1003816100381610038161003816100381610038161003816100381610038161003816
119887119886119887211988611988721198861198872119886119887119886119887
119887119886119887119886119887211988611988721198861198872119886119887
21119860 161280 119887119886119887211988611988721198861198872119886119887119886119887119886119887
21119861 161280 119887119886119887211988611988721198861198872119886119887
2411986080640 119887119886119887119886119887
2119886119887
80640 1198871198861198872119886119887119886119887
Table 34
119883119862
|O| 119892
2119860 315 ((119886119887)3119886)2119886((119886119887)
3119886)4119886(119886119887)
3((119886119887)3119886)2119886
2119861 1800 1198861198871198861198871198861198872
3119860 200 119887211988611988721198861198871198861198872119886119887119886119887
3119862 16800 11988611988721198861198872119886119887119886119887119886
41198603150 119887119886119887
21198861198871198861198872119886
630 1198871198861198872119886119887119886119887119886119887119886119887119886119887
2119886
4119862 12600 1198871198861198872119886119887119886
5119861 20160 119886119887119886
6119860 12600 1198861198871198861198871198861198872119886119887
6119864 100800 119887119886
7119860 100800 119887
811986025200 119886119887119886119887119886119887
25200 119887119886119887119886119887119886119887
12119860 50400 119886
Table 35
119883119862
|O| 119892
2119860 495 (119886119887)5119886 (119886119887)
2(119886 (119886119887)
5)4
(119886119887)5
2119861 792 11988721198861198871198861198872119886119887119886119887119886119887
2119886119887119886119887
3119860 264 119886(119886119887)9((119886119887)9119886)4119886(119886119887)
9((119886119887)9119886)2
31198615280 119886(119886119887)
6((119886119887)3119886)4
1760 (119886((119886119887)3119886)2)4119886
3119862 2640 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886
4119860
2970 11988711988611988711988611988721198861198871198861198872119886119887119886
495
495
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886
4119861 11880 119887119886119887211988611988721198861198871198861198872119886119887119886
4119862 11880 1198872119886119887
5119860 19008 119887119886119887119886
5119861 47520 1198861198872119886119887119886119887119886
6119860
11880 119886119887119886
3960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198871198861198872119886
11988611988711988611988721198861198872119886
611986115840 119886119887
2119886119887119886119887
15840 1198861198872119886119887119886119887119886119887119886119887119886119887
611986215840 119887
21198861198872119886119887119886
15840 1198872119886119887119886119887119886119887119886119887119886119887
21198861198872
611986347520 119886119887
2119886119887119886119887119886119887119886
15840 1198861198872119886119887119886119887119886119887
2119886119887119886
Algebra 11
Table 35 Continued
119883119862
|O| 119892
6119864 15840 119886119887119886119887119886119887119886119887211988611988711988611988721198861198872119886119887
7119860 47520 1198871198861198872119886119887119886
811986147520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886
119886119887119886119887119886119887
911986095040 119887119886119887119886119887119886119887
31680 119887119886119887119886119887211988611988721198861198871198861198872119886
911986195040 119887
2119886119887119886119887119886
31680 1198861198872119886119887119886119887119886119887119886119887119886119887119886
10119860 95040 119886119887
1011986147520 119887119886119887119886119887
21198861198872119886
47520 11988711988611988721198861198872119886119887119886
11119860 190080 119886
12119860 47520 1198872119886119887119886
12119861
47520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198872119886119887119886119887119886119887
2
119886119887119886119887119886119887119886119887119886
15840
15840
1003816100381610038161003816100381610038161003816100381610038161003816
11988721198861198872119886119887119886119887
119887119886119887119886119887119886
1211986247520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198872119886119887119886
1198861198871198861198871198861198871198861198871198861198871198861198872
12119864
47520 11988721198861198872119886
47520 119886119887119886119887
47520 119887119886119887119886119887119886119887119886119887
47520 1198872119886119887119886119887119886119887119886
1411986047520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198871198861198872119886119887119886119887
119887
15119862 190080 119886119887119886119887119886
1811986095040 119887119886119887
2119886119887119886119887119886
95040 11988611988721198861198872119886119887119886119887
1811986195040 119887
21198861198872119886119887119886119887119886
95040 1198871198861198871198861198871198861198872119886
2011986095040
95040
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198872119886119887119886
119887119886
Table 36
119883119862
|O| 119892
2119860 630 119886119887211988621198872119886
3119861 8960 11988611988721198862
3119862 1920 11988621198871198862
Table 37
119883119862
|O| 119892
4119861 30240 1198861198872119886
5119861 48384 1198861198871198862
6119862 80640 119886
Table 38
119883119862
|O| 119892
2119860 495 119887119886119887119886211988711988621198871198861198872119886119887
2119861396 119886119887
311988611988721198862119887119886
396 11988621198873119886119887211988621198871198862
31198601320 119887119886
211988711988621198872119886119887
440 11988611988721198861198871198861198872119886119887119886119887
31198612640 119886
211988721198861198871198861198873119886
1584 119886119887119886119887119886211988731198862
3119862 7920 11988711988611988721198861198872119886119887
4119860 2970 1198871198861198871198861198871198861198872119886119887119886
4119861
11880 1198862119887
5940 1198872
2970 1198871198861198871198861198872119886119887
51198609504 119887
21198861198871198862
9504 119887119886211988731198862119887119886
5119861 47520 11988621198872119886
611986011880 119887119886
211988721198861198872119886
3960 119886119887211988621198871198861198871198862119887119886
611986115840 119886
2
15840 119886
6119862
23760 1198871198861198871198861198873119886
23760 11988711988621198872119886119887119886
15840 1198871198861198873119886119887119886
611986423760 119887
2119886
23760 11988621198872
711986095040 119887119886
2
31680 1198871198861198872119886119887119886
811986023760 119887119886119887119886119887
2119886
23760 1198872119886119887119886119887119886119887
811986123760 119886119887119886119887
2119886
23760 11988721198862119887119886
3142 119883 = 2119861 119905 = 119886 See Table 33
3143 119883 = 2119862 119905 = 119888 See Table 34
3144 119883 = 2119863 119905 = (119888(119888119889)12)11 See Table 35
315 119870 cong 1198621199003
3151 119883 = 2119860 119905 = 1198872 See Tables 36 and 37
12 Algebra
Table 39
119883119862
|O| 119892
8119862
23760 11988611988721198861198872
23760 1198872119886211988721198862
23760 1198872119886119887119886
23760 11988711988621198871198861198873
911986031680 119887119886119887
21198861198873
31680 119886119887119886119887119886119887119886119887
9119861 63360 1198861198871198861198871198861198872
1011986047520 119886119887119886
21198872
47520 1198871198861198871198861198872
1011986147520 119886119887119886
47520 119886211988731198862
1211986023760 119886
21198871198862
23760 1198861198873119886
12119861
47520 119887
47520 1198861198871198861198871198862
47520 1198871198861198872119886119887
1411986095040 119887119886119887119886
95040 1198861198872119886119887119886
1511986095040 119886119887
2
95040 119886119887119886119887
15119861 190080 1198861198871198861198872
1811986095040 119886119887
2119886119887
95040 119887119886119887119886119887
21119860 190080 11988711988621198871198862
2411986095040 119886119887
2119886
95040 119886211988721198862
2411986195040 119886
2119887119886
95040 119887119886
3011986095040 119886119887119886
2
95040 11988611988731198862
3152 119883 = 2119861 119905 = ((1198862119887)3119886119887)5 See Tables 38 and 39
316 119870 cong 1198621199002
3161 119883 = 2119860 119905 = 119886 See Table 40
3162 119883 = 2119861 119905 = 119886(119886119887119886119887119886)2119886(119886119887119886119887119886)7 See Table 41
3163 119883 = 2119862 119905 = (119887minus1(1198861198872)2119886119887minus1(1198861198872)4)5 See Table 42
Table 40
119883119862
|O| 119892
2119861 1260 1198871198861198872119886119887
2119862 1008 11988731198861198871198861198873
3119861 14336 119887119886119887
4119862 40320 119887
Table 41
119883119862
|O| 119892
2119860 15 (1198874119886)2119887311988611988721198861198874(119887119886119887)
2119886(119887119886119887)
711988611988711988611988731198861198872(119886119887)2
21198611680 119887
31198861198871198861198873
210 11988741198861198874119886119887119886119887
2119862 2520 (119886119887)5119886(119886119887)
2119886(119886119887)
3
3119861 35840 1198872119886119887
4119860 1920 11988711988611988731198861198872119886119887119886119887
4119862 20160 1198871198861198872119886119887
4119864 161280 119887119886119887
411986513440 119887119886119887119886119887119886119887119886119887
13440 119887311988611988711988611988721198861198872
5119861 344064 1198872
6119864 430080 119887
Table 42
119883119862
|O| 119892
2119860240 119886119887
4119886119887211988611988711988611988741198861198873119886119887311988611988741198861198873119886119887119886
45 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887411988611988741198861198872119886
2119861
720 1198872119886
240 11988741198861198874119886119887311988611988741198861198874119886119887211988611988731198861198872119886
180 119886119887211988611988731198861198872119886119887119886119887311988611988731198861198872
30 1198861198871198861198874119886
2119862
1440 1198861198871198861198873119886119887311988611988721198861198871198861198872119886
1440 119886119887311988611988741198861198873119886119887211988611988721198861198874119886
1440 1198872119886119887119886119887119886119887
311988611988741198861198873
1152 11988611988711988611988731198861198873119886119887311988611988741198861198872119886
360 11988611988711988611988711988611988731198861198872119886119887119886119887311988611988731198861198872
3119860 10240 1198861198874119886119887211988611988711988611988721198861198871198861198872119886
3119861
15360 11988611988731198861198872119886
7680 11988731198861198872
1024 1198861198872119886119887211988611988731198861198873119886
41198603840 119887
411988611988741198861198873119886119887411988611988741198861198874
1440 1198861198871198861198872119886119887211988611988741198861198872
4119861
11520 1198871198861198874119886119887411988611988731198861198873119886
5760 119886119887211988611988711988611988721198861198871198861198873119886
5760 11988611988721198861198874119886119887311988611988741198861198873119886
1440 11988611988741198861198871198861198872119886119887411988611988721198861198871198861198873119886
1440 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887119886
720 119886119887119886119887411988611988721198861198872119886119887211988611988731198861198873
720 1198872119886119887411988611988721198861198872119886119887411988611988731198861198874119886
Algebra 13
Table 42 Continued
119883119862
|O| 119892
4119862
11520 11988611988721198861198873119886
11520 1198861198873119886119887119886119887311988611988721198861198872
5760 119887311988611988741198861198872119886
5760 11988611988731198861198871198861198872
2880 11988611988711988611988741198861198872119886
1440 119887411988611988741198861198873119886119887211988611988721198861198873119886
1440 1198861198872119886119887311988611988731198861198872119886119887119886119887
720 11988611988711988611988711988611988711988611988721198861198871198861198873119886
720 119886119887311988611988741198861198872119886
4119863
23040 1198871198861198873119886119887119886119887211988611988731198861198872
11520 119886119887119886119887311988611988731198861198871198861198873
5760 119886119887411988611988711988611988741198861198872119886
2880 11988611988711988611988711988611988741198861198872119886119887411988611988741198861198873119886
2880 1198861198872119886119887119886119887119886119887
311988611988711988611988741198861198874119886
4119864
23040 1198871198861198871198861198871198861198872
23040 119886119887311988611988721198861198872
11520 1198861198873119886119887211988611988741198861198872
11520 119887311988611988711988611988731198861198872119886
5760 11988731198861198874119886
5760 1198861198871198861198872
5760 1198874119886119887311988611988721198861198871198861198873119886
4119865
23040 119886119887119886119887211988611988721198861198872
23040 1198873119886119887311988611988731198861198874119886
11520 119887411988611988741198861198872119886119887411988611988731198861198872119886
11520 119886119887311988611988741198861198873119886119887211988611988721198861198872
5760 1198872119886119887411988611988721198861198872119886119887211988611988731198861198872119886
5760 1198872119886119887211988611988731198861198873119886119887411988611988721198861198873119886
5760 1198861198873119886119887211988611988741198861198872119886119887311988611988731198861198872119886
5760 1198861198874119886119887119886119887119886119887
3119886119887211988611988721198861198873
5760 119886119887211988611988731198861198871198861198872119886119887211988611988731198861198873
5760 1198874119886119887411988611988721198861198872119886119887211988611988731198861198872119886
2880 119887311988611988721198861198871198861198872
2880 1198873119886119887411988611988731198861198872
1920 11988611988711988611988741198861198872119886119887211988611988741198861198873
1920 1198871198861198873119886119887311988611988711988611988721198861198874
4119866
46080 1198873119886119887311988611988721198861198873119886
46080 1198861198872119886119887311988611988721198861198872
23040 1198861198871198861198872119886119887211988611988731198861198872119886
23040 11988611988731198861198872119886119887311988611988731198861198874119886
5119860 73728 119886119887211988611988731198861198871198861198872119886
5119861184320 119887
21198861198872119886
122880 119887211988611988731198861198872
Table 42 Continued
119883119862
|O| 119892
6119860 92160 1198871198861198873119886119887411988611988731198861198872119886
6119861 122880 119887211988611988741198861198872119886
6119862
46080 1198872119886119887311988611988731198861198872119886
46080 119887311988611988711988611988721198861198872
15360 1198874119886119887211988611988731198861198873119886
15360 119886119887211988611988721198861198873119886119887
6119863
92160 119887411988611988711988611988721198861198874119886
23040 11988741198861198873119886119887411988611988721198861198872
15360 11988611988721198861198872119886119887119886
15360 119886119887411988611988731198861198873119886
6119864
92160 1198871198861198871198861198872
92160 1198871198861198874119886
46080 1198873119886119887211988611988731198861198872119886
46080 11988711988611988721198861198873119886
46080 11988721198861198871198861198874119886
46080 11988611988711988611988741198861198873
15360 119886119887211988611988711988611988741198861198871198861198872119886
6119865
92160 1198861198873119886119887119886
92160 11988611988741198861198872119886
46080 119887311988611988721198861198871198861198872119886
46080 1198861198874119886119887411988611988731198861198873
46080 1198861198873119886119887411988611988731198861198872
46080 11988721198861198872119886119887119886119887119886
7119860 737280 11988741198861198872
811986092160 119887
2119886119887119886
92160 11988611988741198861198873
8119861
92160 119886119887411988611988721198861198872119886119887119886
92160 119886119887411988611988731198861198873119886119887119886
46080 11988611988711988611988731198861198872119886
46080 119886119887311988611988721198861198874119886
23040 119887119886119887311988611988731198861198871198861198873
23040 11988721198861198874119886119887211988611988721198861198874
8119862
92160 1198873119886119887311988611988731198861198874
92160 1198872
92160 1198873
92160 119887119886119887211988611988721198861198872
8119863
92160 119887311988611988731198861198872
92160 11988611988721198861198873119886119887311988611988741198861198872
92160 119887411988611988731198861198873119886
92160 11988711988611988721198861198871198861198873119886
92160 119887311988611988721198861198872
14 Algebra
Table 42 Continued
119883119862
|O| 119892
8119864
92160 1198861198873119886
92160 1198861198872119886
46080 119886119887311988611988711988611988741198861198872
46080 119887311988611988711988611988741198861198872119886
46080 119886119887211988611988721198861198871198861198873
46080 1198872119886119887119886119887119886
46080 1198872119886119887411988611988731198861198873119886
46080 1198872119886119887211988611988731198861198873119886
46080 1198861198872119886119887211988611988731198861198873
46080 119886119887411988611988741198861198873
23040 1198861198874119886119887211988611988721198861198871198861198873
23040 11988741198861198874119886119887211988611988741198861198873
8119865
184320 11988611988741198861198871198861198873119886
184320 11988611988721198861198874119886119887119886
92160 119886119887119886119887311988611988711988611988721198861198872
92160 11988731198861198874119886119887411988611988731198861198872119886
9119860737280 119886119887
41198861198871198861198872
245760 119887211988611988741198861198872
10119860 368640 119887211988611988731198861198872119886
10119861368640 119886119887
21198861198872
368640 11988731198861198873119886
10119862368640 119886119887
4
368640 11988611988711988611988731198861198871198861198873119886
11119860 1474560 11988721198861198873119886
12119860122880 119887119886119887
21198861198871198861198874119886
122880 119887411988611988711988611988741198861198873
12119861368640 119887
211988611988711988611988731198861198872
184320 1198861198874119886119887211988611988741198861198872
12119862368640 119886119887
2119886119887211988611988731198861198872
184320 119886119887411988611988711988611988721198861198872
12119863
184320 119887411988611988711988611988721198861198872
184320 11988611988711988611988711988611988731198861198873119886
92160 119886119887411988611988721198861198873119886
92160 11988611988721198861198873119886119887119886
12119864368640 119887
311988611988721198861198874
368640 11988711988611988731198861198872
12119865
184320 11988721198861198871198861198872119886
184320 11988611988731198861198874119886
184320 11988611988711988611988731198861198873
184320 1198861198871198861198872119886
92160 1198861198872119886119887119886
92160 11988611988741198861198873119886
12119866 368640 11988711988611988741198861198872119886
Table 42 Continued
119883119862
|O| 119892
12119867
184320 1198874119886119887311988611988731198861198872
184320 119886119887311988611988721198861198873119886
184320 11988611988711988611988711988611988741198861198872119886
184320 1198861198874119886119887311988611988741198861198872119886
184320 119887311988611988721198861198872119886119887
184320 119886119887211988611988731198861198872119886
14119860 737280 1198861198874119886
15119860737280 119887
4119886
737280 1198871198861198872119886
16119860
368640 11988611988731198861198872
368640 11988611988721198861198871198861198874
368640 11988731198861198872119886
368640 11988711988611988741198861198873119886
18119860 737280 11988731198861198873
20119860737280 119886119887
41198861198872
737280 1198873119886119887119886
20119861737280 119886119887
41198861198874119886
737280 119886119887119886119887119886
24119860737280 119886119887
2
737280 1198873119886
24119861737280 119887
4
737280 119887
28119860737280 119886119887119886119887
31198861198872
737280 119886119887411988611988721198861198872
30119860737280 119887
21198861198874
737280 1198871198861198873
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R H Bing ldquoA homeomorphism between the 3-sphere andthe sum of two solid horned spheresrdquo Annals of MathematicsSecond Series vol 56 pp 354ndash362 1952
[2] D Montgomery and L Zippin ldquoExamples of transformationgroupsrdquo Proceedings of the American Mathematical Society vol5 pp 460ndash465 1954
[3] J W Morgan and H Bass Eds The Smith Conjecture vol 112of Pure and Applied Mathematics Academic Press OrlandoFla USA 1984 Papers presented at the symposium held atColumbia University New York NY USA 1979
[4] C M Edwards and G T Ruttimann ldquoInvolutive and Peircegradings in JBWlowast-triplesrdquo Communications in Algebra vol 31no 6 pp 2819ndash2848 2003
Algebra 15
[5] J Gornicki and B E Rhoades ldquoA general fixed point theoremfor involutionsrdquo Indian Journal of Pure and Applied Mathemat-ics vol 27 no 1 pp 13ndash23 1996
[6] S Montgomery Fixed Rings of Finite Automorphism Groupsof Associative Rings vol 818 of Lecture Notes in MathematicsSpringer Berlin Germany 1980
[7] T A Springer ldquoThe classification of involutions of simplealgebraic groupsrdquo Journal of the Faculty of Science University ofTokyo Section IA Mathematics vol 34 no 3 pp 655ndash670 1987
[8] T A Springer ldquoSome results on algebraic group s with invo-lutionsrdquo in Algebraic Groups and Related Topics (KyotoNagoya1983) vol 6 of Advanced Studies in Pure Mathematics pp 525ndash543 North-Holland Amsterdam The Netherlands 1985
[9] W Feit and J GThompson ldquoSolvability of groups of odd orderrdquoPacific Journal of Mathematics vol 13 pp 775ndash1029 1963
[10] R Brauer and K A Fowler ldquoOn groups of even orderrdquo Annalsof Mathematics vol 62 no 3 pp 565ndash583 1955
[11] D Gorenstein The Classification of Finite Ssimple Groups Vol1 Groups of Noncharacteristic 2 Type The University Series inMathematics Plenum Press New York NY USA 1983
[12] D Gorenstein R Lyons and R Solomon The Classification ofthe Finite Simple Groups vol 40 of Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 1994
[13] C Bates and P Rowley ldquoInvolutions in Conwayrsquos largest simplegrouprdquo LMS Journal of Computation andMathematics vol 7 pp337ndash351 2004
[14] P Rowley and P Taylor ldquoPoint-line collinearity graphs of twosporadic minimal parabolic geometriesrdquo Journal of Algebra vol331 pp 304ndash310 2011
[15] P Rowley and P Taylor ldquoInvolutions in Jankorsquos simple group 1198694rdquoLMS Journal of Computation and Mathematics vol 14 pp 238ndash253 2011
[16] P Taylor ldquoInvolutions in Fischerrsquos sporadic groupsrdquo preprinthttpeprintsmamanacuk1622
[17] The GAP Group GAP-Groups Algorithms Programming Ver-sion 43 2002 httpwwwgap-systemorg
[18] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I The user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997
[19] R Wilson P Walsh J Tripp et al ldquoAtlas o f finite grouprepresentationsrdquo httpbrauermathsqmulacukAtlasv3
[20] J H Conway R T Curtis S P Norton R A Parker and R AWilson Atlas of Finite Groups Clarendon Oxford UK 1985
[21] R AWilson ldquoStandard generators for sporadic simple groupsrdquoJournal of Algebra vol 184 no 2 pp 505ndash515 1996
[22] D Gorenstein Finite Groups Chelsea Publishing New YorkNY USA 2nd edition 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Algebra
Table 6
119883119862
|O| 119892
211986015 119887119886119887
2119886119887119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
15 119887119886119887211988611988721198861198871198861198872119886119887119886119887119886119887
2119886119887119886119887
311986060 119887119886119887
2119886119887119886119887119886119887
2119886119887
12 1198872119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
511986060 119887119886119887119886119887
21198861198872119886119887119886119887
20 1198872119886119887119886119887119886119887
21198861198872119886119887
511986160 119887119886119887119886119887
21198861198871198861198872119886119887119886119887
20 11988721198861198871198861198872119886119887119886119887
611986060 119887119886119887
2119886119887119886119887
60 1198871198861198871198861198872119886119887
7119860 120 11988711988611988721198861198872119886119887119886119887
1011986060 119887
2119886119887119886119887119886119887
60 1198871198861198871198861198871198861198872
1011986160 119887
2119886119887119886119887
60 1198871198861198871198861198872
11119860 120 1198871198861198871198861198872119886119887119886119887
15119860 120 11988721198861198872119886119887119886119887
15119861 120 119887119886119887
19119860 120 119887
19119861 120 1198872119886119887
19119862 120 11988721198861198871198861198872119886119887
Table 7
119883119862
|O| 119892
2119860
24 1198871198861198873119886119887
12 (119887119886)21198872119886119887119886119887
8 11988721198861198871198861198872(119886119887)3
6 1198873(119886119887)2(1198861198873)2119886119887
3119860 192 119887119886119887
4119860
48 1198871198861198872119886119887
48 1198873119886119887119886119887
48 1198871198861198871198861198873
411986196 119887
2119886119887119886119887
96 1198871198861198871198861198872
5119860 384 1198872
6119860 192 119887
Table 8
119883119862
|O| 119892
211986042 119887
7 (1198872119886)2119887119886
3119860 112 119886
4119860 168 119886119887119886119887119886
Table 9
119883119862
|O| 119892
211986020 119887
2
5 11988711988611988721198861198873
3119860 80 119887
4119860 80 119887119886119887119886119887119886
411986140 119887119886119887
2119886
40 1198871198861198872119886119887
5119860 320 119886119887
611986080
80
1003816100381610038161003816100381610038161003816100381610038161003816
119886
1198861198872119886
11119860 320 1198872119886
11119861 320 119887119886
Table 10
119883119862
|O| 119892
2119860 10 119887119886
3119861 160 119887
4119860 80 119886
5119860 32 1198861198872119886119887119886
5119861 32 1198871198861198872
362 119883 = 2119861 119905 = 119886 See Table 11
363 119883 = 2119862 119905 = 119888 See Table 12
37 119870 cong 11987223
371119883 = 2119860 119905 = 119886 See Table 13
38 119870 cong 119867119878
381 119883 = 2119860 119905 = 119886 See Table 14
382 119883 = 2119861 119905 = ((119886119887)2119886)5 See Tables 15 and 16
383 119883 = 2119862 119905 = 119888 See Table 17
384 119883 = 2119863 119905 = ((119889119888)21198894119888119889)5 See Tables 18 and 19
39 119870 cong 1198693
391119883 = 2119860 119905 = 119886 See Table 20
392119883 = 2119861 119905 = 119888 See Table 21
310 119870 cong11987224
3101 119883 = 2119860 119905 = ((119886119887)31198861198872)4 See Tables 22 and 23
Algebra 5
Table 11
119883119862
|O| 119892
2119860 15 119887(119887119886)2119887(119887119886)
2119887(119887119886)
2
2119861
15
15
1003816100381610038161003816100381610038161003816100381610038161003816
((119887119886)2119887)3(119887119886)3
119887(119887119886)2(119887(119887119886)
31198872119886)2
1
1
1003816100381610038161003816100381610038161003816100381610038161003816
(119887119886)21198872119886(119887(119887119886)
2)21198872119886119887(119887119886)
3
1198872119886119887(119887119886)
3119887(119887119886)
2119887(119887119886)
3(1198872119886)2
3119860 24 119887(119887119886)2119887(119887119886)
2
3119861 60 119887(119887119886)3119887(119887119886)
2119887(119887119886)
3119887(119887119886)
4119860 60 (119887119886)119887(119887119886)2
5119860 60 (119887119886)2119887(119887119886)
2119887(119887119886)
3119887(119887119886)119887(119887119886)
2
5119861 60 (119887119886)119887(119887119886)119887(119887119886)3119887(119887119886)
2119887(119887119886)
3
5119862120 119887(119887119886)
3119887(119887119886)
3
24 119887(119887119886)119887(119887119886)2119887(119887119886)
3
5119863120 (119887119886)119887(119887119886)
2119887(119887119886)
3
24 (119887119886)2119887(119887119886)
2119887(119887119886)119887(119887119886)
6119860 120 (119887119886)3
611986160
60
1003816100381610038161003816100381610038161003816100381610038161003816
(119887119886)2119887(119887119886)
3
(119887119886)3119887(119887119886)
4
7119860 240 119887(119887119886)4
1011986060 119887(119887119886)119887(119887119886)
2119887(119887119886)
2
60 119887(119887119886)2119887(119887119886)
3
1011986160 (119887119886)
2119887(119887119886)
2119887(119887119886)
60 (119887119886)119887(119887119886)2119887(119887119886)119887(119887119886)
10119862120 (119887119886)119887(119887119886)119887(119887119886)
120 (119887119886)2119887(119887119886)
10119863120 119887(119887119886)
3
120 119887(119887119886)119887(119887119886)2
12119860120
120
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119887119886)
(119887119886)
15119860 240 (119887119886)119887(119887119886)119887(119887119886)2119887(119887119886)
2
15119861 240 (119887119886)119887(119887119886)2119887(119887119886)119887(119887119886)
2
Table 12
119883119862
|O| 119892
2119860 21 (1198871198861198872119886)2(119887119886)3
2119861 28 1198871198861198872
311986014
14
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)3(1198861198872)2119886119887
(1198872119886119887119886)2119887119886119887
3119861 168 119887119886119887119886
411986084 119886119887119886119887
2119886119887
42 1198872119886119887119886119887
6119860
84
84
1003816100381610038161003816100381610038161003816100381610038161003816
1198872
119887
42
42
1003816100381610038161003816100381610038161003816100381610038161003816
119886119887119886119887119886119887119886
1198861198871198861198872119886119887119886119887
6119861 168 119886119887
7119860 336 119887119886119887
8119860168 119886119887
2119886
168 119886119887119886
12119860168
168
1003816100381610038161003816100381610038161003816100381610038161003816
119887119886
119886
Table 13
119883119862
|O| 119892
211986084 119887119886119887
3119886119887
14 11988731198861198872119886119887
3119860 448 1198872
4119860
336 119887119886119887119886119887
336 119887119886119887119886119887119886119887
336 1198871198861198871198861198871198861198872
5119860 896 119887119886119887
6119860 1344 119887
Table 14
119883119862
|O| 119892
211986080 119887
211988611988731198861198872
30 119887211988611988711988611988721198861198871198861198872
3119860 640 119887119886119887
4119861 480 1198871198861198873119886119887
4119862 960 1198872119886119887
5119861 128 1198871198861198872119886119887
5119862 1536 119887
6119861 1920 1198872
Table 15
119883119862
|O| 119892
211986045 119887119886119887
3119886119887
30 11988741198861198874119886119887119886119887
211986136
36
1003816100381610038161003816100381610038161003816100381610038161003816
119887211988611988731198861198872
1198871198861198871198861198871198861198873119886
3119860360 119887
31198861198872119886119887119886119887
24 1198861198872119886119887211988611988711988611988721198861198872119886
4119861 180 119886
4119862
360 11988711988611988711988611988741198861198872119886
180
180
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198871198861198872
1198872119886119887119886119887119886119887
5119860144
144
1003816100381610038161003816100381610038161003816100381610038161003816
11988711988611988741198861198873119886119887
11988711988611988731198861198874119886119887
5119861 720 1198871198861198871198861198872119886
Table 16
119883119862
|O| 119892
6119861
720 119887119886119887
360 1198871198861198872119886
360 1198861198872119886119887
7119860 2880 119887
8119861720 119887
41198861198872119886
720 1198861198873119886119887
8119862720 119887119886119887
3119886
720 11988611988721198861198874
10119860720
720
1003816100381610038161003816100381610038161003816100381610038161003816
1198872
1198873
10119861720
720
1003816100381610038161003816100381610038161003816100381610038161003816
119886119887119886
119887119886
15119860 2880 1198872119886
6 Algebra
Table 17
119883119862
|O| 119892
2119860 105 119886
2119861 28 119887119886
3119860 336 1198872
4119861 630 119887
Table 18
119883119862
|O| 119892
211986060 119886119887
31198861198871198861198872119886
5 11988611988731198861198872119886
2119861 60 1198872119886
3119860240 119886
160 11988611988731198861198871198861198872119886119887119886
4119860 120 1198871198861198873119886119887119886
4119861
240 119886119887
120 1198871198861198872119886119887119886119887
30 1198861198872119886
4119862120 119886119887
21198861198872119886
120 1198861198873119886119887119886
5119860192 119886119887119886119887
3
192 11988711988611988721198861198873
5119861 320 1198874119886119887
5119862 1920 11988721198861198872
6119860480 119886119887
2119886119887
480 1198871198861198872119886
Table 19
119883119862
|O| 119892
6119861480 119887
4119886119887119886119887
480 11988721198861198872119886
7119860 1920 119887
8119860480 119887119886119887119886119887
2
480 11988711988611988721198861198872119886119887
10119860960 119887
211988611988721198861198872119886
960 119887119886119887
10119861 960 1198861198872
11119860 1920 1198872119886119887119886
11119861 1920 119887119886119887119886119887
12119860960 119887119886
960 1198873
15119860 1920 1198871198861198873
20119860960 119887119886119887119886
960 1198872119886119887
20119861960 119887
21198861198873119886
960 1198874119886119887119886
Table 20
119883119862
|O| 119892
2119860120 119887(119887119886)
3119887(119887119886)119887(119887119886)
5
10 (119887119886)119887(119887119886)4119887(119887119886)
2
3119860 192 119887(119887119886)6
3119861 640 (119887119886)119887(119887119886)2
4119860480 119887(119887119886)
5119887(119887119886)
80 (119887119886)3119887(119887119886)
3119887(119887119886)
4
5119860960 (119887119886)
3119887(119887119886)
3119887(119887119886)
2
320 (119887119886)2119887(119887119886)
2
5119861960 (119887119886)119887(119887119886)
3119887(119887119886)
4
320 (119887119886)119887(119887119886)3
6119860 960 (119887119886)119887(119887119886)6
8119860960 (119887119886)
4119887(119887119886)
2
960 (119887119886)119887(119887119886)5
9119860 1920 119887(119887119886)2119887(119887119886)
2
9119861 1920 (119887119886)
9119862 1920 119887(119887119886)2
10119860960 (119887119886)
4119887(119887119886)
960 119887(119887119886)4119887(119887119886)119887(119887119886)
10119861960 119887(119887119886)119887(119887119886)119887(119887119886)119887(119887119886)
2
960 119887(119887119886)5
12119860960
960
1003816100381610038161003816100381610038161003816100381610038161003816
(119887119886)2119887(119887119886)
4
(119887119886)6
15119860 1920 (119887119886)2119887(119887119886)
15119861 1920 119887(119887119886)3
17119860 1920 (119887119886)2
17119861 1920 (119887119886)2119887(119887119886)
3
Table 21
119883119862
|O| 119892
2119860 153 1198861198872119886119887119886119887119886119887
3119860102
102
1003816100381610038161003816100381610038161003816100381610038161003816
1198872
119887
3119861 272 1198861198871198861198872119886
4119860612 119887119886119887119886119887119886119887
2119886
306 1198871198861198872119886119887119886119887119886
6119860
612
612
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198872119886119887119886
1198871198861198871198861198872119886
306
306
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198872119886119887119886119887119886
119887119886119887119886
81198601224 119886119887119886119887119886119887
1224 1198872119886119887119886119887119886
9119860 816 11988721198861198871198861198872119886119887
9119861 816 1198871198861198871198861198872119886119887
9119862 816 1198871198861198872
121198601224
1224
1003816100381610038161003816100381610038161003816100381610038161003816
119886119887
1198861198872119886119887119886
17119860 2448 119887119886
17119861 2448 119886
19119860 2448 1198861198872119886119887
19119861 2448 1198872119886119887119886
Algebra 7
Table 22
119883119862
|O| 119892
2119860
168 1198872119886119887119886119887119886119887
2119886119887
14 119886119887119886
14 1198861198871198861198872119886119887
2119861 84 1198872119886119887
3119860 896 119887
3119861 128 119887211988611988721198861198871198861198872119886119887119886119887
Table 23
119883119862
|O| 119892
4119860112 119887119886119887119886119887119886119887
112 1198872119886119887211988611988721198861198872
4119861
1344 1198861198872119886119887119886
672 11988611988721198861198871198861198872119886119887
672 11988721198861198871198861198872119886119887119886
5119860 1792 119887119886119887119886
6119860 5376 119886
Table 24
119883119862
|O| 119892
211986060 119887
2119886119887119886119887211988611988721198861198871198861198872119886119887119886119887119886119887
2119886119887
15 11988711988611988721198861198872119886119887119886119887119886119887
2
2119861
120 119887119886119887211988611988711988611988721198861198871198861198872
80 119887211988611988711988611988721198861198871198861198872119886119887
2 (119887119886)8(1198872119886)21198871198861198872(119886119887)2(1198861198872)2(119886119887)3(1198861198872119886119887)2119887
3119860 320 119887211988611988721198861198871198861198872119886119887119886119887119886119887
3119861 960 11988721198861198871198861198872119886119887
4119860120 119887119886119887
21198861198872
120 1198871198861198871198861198872
4119861
480 1198872119886119887119886119887119886119887
2119886119887
240 1198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
240 1198872119886119887119886119887119886119887119886119887119886119887119886119887
4119862
960 1198871198861198872119886119887
960 11988721198861198871198861198872
160 1198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
21198861198872
160 1198872119886119887119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
5119860 1920 119887119886119887
6119860 1920 1198871198861198871198861198871198861198872
61198611920 119887119886119887
2
1920 1198872119886119887
101198601920 119887
21198861198871198861198872119886119887119886119887
1920 1198872119886119887119886119887119886119887119886119887
2
11119860 7680 119887119886119887119886119887
121198613840 119887
3840 1198872
3102 119883 = 2119861 119905 = 119886 See Table 24
311 119870 cong 119872119888119871
3111119883 = 2119860 119905 = 119886 See Table 25
3112119883 = 2119861 119905 = 119888 See Table 26
Table 25
119883119862
|O| 119892
2119860 210 119887211988611988731198861198872
3119861 2240 1198871198861198871198861198872
4119860 5040 1198872119886119887
5119861 8064 119887
6119861 6720 1198872
Table 26
119883119862
|O| 119892
2119860 165 11988611988721198861198871198861198872119886119887
3119860 220 1198872119886119887311988611988721198861198871198861198872119886
3119861660 119886119887
21198861198872
220 1198872119886119887119886119887119886119887
3
41198601980 119887
21198861198873119886119887119886119887119886
990 119886
5119860792 119887
211988611988721198861198873
792 1198872119886119887119886119887119886119887
2
5119861 7920 1198872
6119860 1980 119886119887
6119861
1980 1198871198861198872119886119887119886119887
1980 1198861198872119886119887
660
660
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198873
11988711988611988731198861198874119886119887
71198603960 119887119886119887119886119887
2119886
3960 1198871198861198872
71198613960 119887119886119887
3119886
3960 1198872119886
81198603960 119887
21198861198872119886
3960 11988611988731198861198873
9119860 2640 1198872119886119887119886
9119861 2640 1198861198871198861198872119886119887
101198603960 119887119886119887119886119887
3960 119886119887119886119887119886
121198603960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198873
1198871198861198873
141198603960 119886119887
21198861198873
3960 1198871198861198872119886119887
141198613960 119887119886119887119886119887119886
3960 1198872119886119887119886119887119886
151198603960 119887119886119887
3960 1198874
151198613960 119886119887119886119887
3960 119887
301198603960 119887119886
3960 1198871198861198872119886
301198613960 119887
41198861198872119886
3960 1198861198872
8 Algebra
Table 27
119883119862
|O| 119892
2119860240 119887119886119887119886119887119886119887
2 1198875(1198861198876)2(119886119887)31198875119886(11988721198861198873)2
2119861 315 119887211988611988711988611988721198861198872119886119887
3119860 1344 1198872119886119887
3119861 960 1198871198861198874119886119887
4119860480 119887
31198861198871198861198872119886119887
480 11988721198861198871198861198873119886119887
41198612520
2520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887
1198874119886119887
5119860 2688 1198873
6119860 13440 119887
Table 28
119883119862
|O| 119892
2119860 84 (119886119887)21198875(119886119887)2
2119861
168
168
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)2
(119886119887)1198874(119886119887)119887
2(119886119887)119887
2(119886119887)119887
4
14
14
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198874(119886119887)119887
4(119886119887)119887
2(119886119887)119887
2
1198872(119886119887)119887
2(119886119887)21198874(119886119887)119887
2(119886119887)119887
2
3119860 896 1198874(119886119887)119887
2
3119861 2688 1198872
4119861
1344
1344
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)119887
2(119886119887)119887
2
1198872(119886119887)119887
4(119886119887)21198874
112
112
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)1198872(119886119887)41198874(119886119887)2
(119886119887)31198874(119886119887)41198872
4119862
1344
1344
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)1198874(119886119887)119887
4
1198874(119886119887)119887
2(119886119887)119887
2(119886119887)31198872
672
672
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198872(119886119887)3
1198872(119886119887)41198872(119886119887)2
672
672
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)41198872(119886119887)2
(119886119887)4119887(119886119887)
2
5119860 5376 (119886119887)3
6119860 5376 (119886119887)1198872(119886119887)119887
2(119886119887)4
61198615376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198872(119886119887)
(119886119887)1198872(119886119887)
7119862 3072 1198872(119886119887)119887
2(119886119887)119887
2(119886119887)2
81198605376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)21198872(119886119887)21198872
(119886119887)1198872(119886119887)21198872(119886119887)
101198605376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)2
1198872(119886119887)119887
2(119886119887)119887
2
1211986110752
10752
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)2
1198874(119886119887)2
15119860 21504 1198872(119886119887)3
17119860 21504 (119886119887)1198872(119886119887)2
17119861 21504 (119886119887)
21119860 21504 (119886119887)1198872(119886119887)119887
2
21119861 21504 1198872(119886119887)
Table 29
119883119862
|O| 119892
2119860105 (119886119887)119887(119886119887)
3119887(119886119887)
3
63 119887611988611988731198861198873119886119887211988611988761198861198872119886119887611988611988761198861198874119886
2119861 315 (1198861198872)2(119887119886)21198872119886
3119860630 (119886119887)119887(119886119887)119887(119886119887)119887
126 119887(119886119887)21198876
3119861 2520 (119886119887)1198876
41198601260 (119886119887)
21198875(119886119887)
1260 (119886119887)119887(119886119887)1198873
4119861
945
945
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)21198872(119886119887)119887
119887(119886119887)119887(119886119887)1198872(119886119887)119887
315
315
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119886119887)5119887
119887(119886119887)31198872(119886119887)119887
41198621890
1890
1003816100381610038161003816100381610038161003816100381610038161003816
1198873(119886119887)2
(119886119887)1198872(119886119887)2119887
5119860 2520 (119886119887)3119887(119886119887)
6119860
3780 (119886119887)31198872(119886119887)2
1890 119887(119886119887)119887(119886119887)
1890 119887(119886119887)119887(119886119887)119887(119886119887)3
1260 119887(119886119887)1198872(119886119887)119887(119886119887)
2
6119861 7560 (119886119887)
71198601890 (119886119887)119887(119886119887)
41198872
90 119887(119886119887)1198873(119886119887)2119887
71198611890 119887
5(119886119887)119887(119886119887)
90 1198875(119886119887)119887
5(119886119887)119887
2(119886119887)119887
5
7119862 2160 119887(119886119887)119887(119886119887)4
71198637560 (119886119887)119887(119886119887)
4
1080 (119886119887)1198873(119886119887)
71198647560 (119886119887)
3119887(119886119887)
2
1080 (119886119887)1198874(119886119887)2119887(119886119887)
8119860
3780
3780
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)2119887(119886119887)
2
(119886119887)4
3780
3780
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)3119887(119886119887)119887(119886119887)119887
(119886119887)119887(119886119887)119887(119886119887)2
10119860 7560 (119886119887)3
121198607560 119887(119886119887)119887
7560 119887(119886119887)1198872
121198617560
7560
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119886119887)2119887
119887
14119860
3780 (119886119887)1198873(119886119887)2
3780 (119886119887)21198872(119886119887)
1890 (119886119887)1198872(119886119887)
1890 119887(119886119887)3119887
14119861
3780 (119886119887)119887(119886119887)2119887(119886119887)
3780 (119886119887)1198874
1890 (119886119887)21198872(119886119887)119887(119886119887)
2
1890 119887(119886119887)119887(119886119887)2
Algebra 9
Table 29 Continued
119883119862
|O| 119892
141198627560 119887(119886119887)
7560 119887(119886119887)3
141198637560 119887
3(119886119887)
7560 119887(119886119887)2
15119860 15120 (119886119887)1198872
21119860 15120 (119886119887)4119887(119886119887)
21119861 15120 (119886119887)119887
21119862 15120 (119886119887)3119887
21119863 15120 1198872
281198607560 119887
2(119886119887)119887
2
7560 (119886119887)2
281198617560 119887
4(119886119887)3
7560 (119886119887)119887(119886119887)2
Table 30
119883119862|O| 119892
2119860
640 11988611988731198861198872119886119887119886
480 11988611988721198861198872119886119887119886119887119886119887119886119887119886119887
21198861198872119886
240 11988611988721198861198873119886119887311988611988731198861198873119886119887211988611988711988611988721198861198873119886119887119886119887119886119887119886119887
31198861198873
30 1198861198872119886119887119886119887119886119887211988611988721198861198873119886119887311988611988711988611988711988611988721198861198873119886119887211988611988721198861198873119886119887211988611988711988611988711988611988731198863119860 10240 119886119887
3119886119887119886119887119886119887119886119887119886119887
2119886
4119860 5120 119886119887119886119887119886119887119886
4119862
7680 119886119887119886119887211988611988721198861198872119886119887119886
3840 119886119887211988611988731198861198871198861198872119886119887119886
3840 119886119887119886119887211988611988711988611988731198861198872119886
411986315360 119886119887119886
7680 1198861198872119886119887211988611988711988611988721198861198871198861198872119886
5119860 12288 11988611988731198861198871198861198872119886119887119886119887119886119887119886
5119861 4096 119886119887119886119887119886
611986061440 119886119887
21198861198871198861198872119886119887119886
30720 119886119887119886119887119886119887119886119887119886119887119886
7119860 61440 119886
811986261440 119886119887119886119887
21198861198872119886
61440 11988611988721198861198872119886119887119886
10119860 61440 11988611988721198861198872119886119887119886119887119886119887119886
12119860122880 1198861198872119886119887119886
13119860 61440 1198861198873119886119887119886
312 119870 cong 119867119890
3121119883 = 2119860 119905 = 119886 See Table 27
3122119883 = 2119861 119905 = (1198861198872)6 See Table 28
3123 119883 = 2119862 119905 = 1198893 See Table 29
313 119870 cong 119877119906
3131 119883 = 2119860 119905 = 1198872 See Table 30
3132 119883 = 2119861 119905 = 119886 See Table 31
314 119870 cong 119878119906119911
3141119883 = 2119860 119905 = ((119886119887)21198861198872)6 See Table 32
Table 31
119883119862 |O| 119892
2119860 455 119887119886119887311988611988721198861198873119886119887
2119861
455 11988711988611988731198861198872119886119887119886119887119886119887119886119887119886119887
455 11988711988611988711988611988711988611988711988611988711988611988721198861198873119886119887
1 1198873119886119887311988611988731198861198872119886119887119886119887119886119887
211988611988721198861198873119886119887119886119887211988611988721198861198872
1 1198871198861198872119886119887119886119887311988611988711988611988721198861198872119886119887119886119887211988611988711988611988731198861198873119886119887119886119887119886119887
31198861198871198861198872119886119887119886119887211988611988731198861198871198861198873
3119860 5824 1198872119886119887119886119887119886119887119886119887119886119887
4119860 3640 (119886119887)10119886(119886119887)15119886((119886119887)
5119886)2
4119862 3640 1198873119886119887119886119887119886119887
21198861198872
4119863 3640 119887311988611988711988611988721198861198873119886119887
51198605824 119887
311988611988711988611988721198861198871198861198872
5824 119887211988611988731198861198872119886119887
5119861 29120 1198872119886119887119886119887
6119860 29120 1198872119886119887119886119887119886119887
2119886119887
7119860
29120 11988721198861198872
29120 119887
29120 1198872
1011986029120 119887119886119887119886119887
2119886119887119886119887
29120 119887119886119887119886119887
1011986129120 119887119886119887119886119887
31198861198873119886119887
29120 11988711988611988731198861198873119886119887119886119887
12119860 58240 1198873119886119887119886119887119886119887
2119886119887
13119860
29120 119887119886119887
29120 119887119886119887119886119887119886119887
29120 11988731198861198872119886119887
1411986029120 119887
311988611988721198861198872119886119887119886119887
29120 11988711988611988721198861198872119886119887119886119887119886119887
1411986129120 119887
21198861198872119886119887119886119887119886119887
29120 11988711988611988711988611988711988611988721198861198872
1411986229120 119887119886119887
2119886119887119886119887119886119887119886119887
2
29120 1198872119886119887119886119887119886119887119886119887
2119886119887
15119860 116480 11988731198861198872119886119887119886119887119886119887
2011986058240 119887
31198861198871198861198872
58240 11988721198861198873119886119887
2611986029120 119887119886119887
2
29120 1198872119886119887
2611986129120 119887119886119887
21198861198872
29120 11988721198861198872119886119887
2611986229120 119887
3119886119887119886119887119886119887
29120 11988711988611988711988611988721198861198872
29119860 116480 1198871198861198872119886119887
29119861 116480 1198873119886119887119886119887
Table 32
119883119862
|O| 119892
2119860360 119886
54 ((119886119887)2119886)4119886((119886119887)
2119886)3
3119861 5120 11988721198861198872119886119887119886119887119886119887119886119887119886119887119886
3119862 9216 119887119886119887119886
4119860 1728 119887211988611988721198861198872119886119887119886119887119886119887
4119862 17280 1198872119886119887
5119861 55296 1198872119886119887119886119887119886
6119863 46080 119887
10 Algebra
Table 33119883119862|O| 119892
2119860 315 1198871198861198872119886119887119886119887119886119887
21198861198872119886119887211988611988721198861198872119886119887119886119887119886119887
2119886119887
2119861
630 1198872119886119887119886119887119886119887119886119887
2119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
21198861198872119886119887119886119887211988611988721198861198872119886119887
560 119887119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
2 (1198871198861198872119886)2(1198872119886)2(119887119886(1198872119886)2)2(1198872119886119887119886)2((1198872119886)21198872119886119887119886)21198871198861198872(119886119887)2
3119860 224 (1198871198861198872119886)3119887(119887119886)3(1198872119886)2(119887119886)5119887(119887119886)4(1198872119886)2119887
3119861 2240 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887
3119862 20160 11988711988611988721198861198871198861198872119886119887
41198602520 119887119886119887119886119887119886119887
2119886119887211988611988721198861198872119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886119887119886119887
4119861
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
2520 11988711988611988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
21198861198872119886119887
1260 119887211988611988721198861198871198861198872119886119887119886119887
4119862
5040 11988711988611988721198861198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
2520 11988711988611988711988611988711988611988721198861198872119886119887119886119887119886119887
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
4119863
10080 11988711988611988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
10080 11988711988611988711988611988721198861198872119886119887119886119887211988611988721198861198872119886119887
1120 (119886119887)4((119886119887)4119886)4119886(119886119887)8((119886119887)4119886)2
1120 (119886(1198872119886)4)2(1198872119886)8119886(119886(1198872119886)4)4(1198872119886)4
5119860 20160 119887119886119887119886119887
5119861 40320 1198872119886119887119886119887119886119887
2119886119887
6119860 10080 1198871198861198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
6119863 20160 11988711988611988721198861198872119886119887
611986440320 119887
2119886119887119886119887119886119887
2119886119887119886119887119886119887
40320 1198872119886119887119886119887211988611988721198861198872119886119887119886119887119886119887
7119860 80640 1198872119886119887119886119887119886119887119886119887
811986020160 119887119886119887
2119886119887119886119887211988611988721198861198872119886119887
20160 1198871198861198872119886119887211988611988721198861198871198861198872119886119887
8119861
20160 1198871198861198872119886119887119886119887119886119887119886119887
211988611988721198861198871198861198872119886119887
20160 11988711988611988711988611988721198861198872119886119887
20160 11988711988611988721198861198872119886119887119886119887
20160 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887119886119887
2119886119887
10080 11988721198861198872119886119887
10080 1198872119886119887119886119887
8119862
20160 11988721198861198871198861198872119886119887119886119887119886119887
20160 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
2119886119887
20160 1198872119886119887211988611988721198861198871198861198872119886119887
20160 1198871198861198872119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
10119860
40320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
119887211988611988711988611988721198861198871198861198872119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886119887
20160 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887
20160 11988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
1011986140320 119887119886119887119886119887119886119887119886119887
2119886119887
40320 1198871198861198872119886119887119886119887119886119887119886119887
11119860 161280 11988721198861198871198861198872119886119887119886119887
1211986040320 119887
2119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
20160 119887211988611988711988611988721198861198872119886119887119886119887119886119887
2119886119887
1211986140320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886119887
11988721198861198872119886119887211988611988721198861198872119886119887
12119862 40320 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
1211986380640 119887119886119887
2119886119887119886119887119886119887
80640 1198871198861198871198861198871198861198872119886119887
13119860 161280 119887119886119887
13119861 161280 11988721198861198872119886119887119886119887
14119860 80640 119887211988611988711988611988721198861198872119886119887119886119887
15119860 161280 119887
15119861 161280 1198872119886119887
15119862 161280 119887211988611988721198861198871198861198872119886119887119886119887119886119887
Table 33 Continued119883119862
|O| 119892
2011986080640
80640
1003816100381610038161003816100381610038161003816100381610038161003816
119887119886119887211988611988721198861198872119886119887119886119887
119887119886119887119886119887211988611988721198861198872119886119887
21119860 161280 119887119886119887211988611988721198861198872119886119887119886119887119886119887
21119861 161280 119887119886119887211988611988721198861198872119886119887
2411986080640 119887119886119887119886119887
2119886119887
80640 1198871198861198872119886119887119886119887
Table 34
119883119862
|O| 119892
2119860 315 ((119886119887)3119886)2119886((119886119887)
3119886)4119886(119886119887)
3((119886119887)3119886)2119886
2119861 1800 1198861198871198861198871198861198872
3119860 200 119887211988611988721198861198871198861198872119886119887119886119887
3119862 16800 11988611988721198861198872119886119887119886119887119886
41198603150 119887119886119887
21198861198871198861198872119886
630 1198871198861198872119886119887119886119887119886119887119886119887119886119887
2119886
4119862 12600 1198871198861198872119886119887119886
5119861 20160 119886119887119886
6119860 12600 1198861198871198861198871198861198872119886119887
6119864 100800 119887119886
7119860 100800 119887
811986025200 119886119887119886119887119886119887
25200 119887119886119887119886119887119886119887
12119860 50400 119886
Table 35
119883119862
|O| 119892
2119860 495 (119886119887)5119886 (119886119887)
2(119886 (119886119887)
5)4
(119886119887)5
2119861 792 11988721198861198871198861198872119886119887119886119887119886119887
2119886119887119886119887
3119860 264 119886(119886119887)9((119886119887)9119886)4119886(119886119887)
9((119886119887)9119886)2
31198615280 119886(119886119887)
6((119886119887)3119886)4
1760 (119886((119886119887)3119886)2)4119886
3119862 2640 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886
4119860
2970 11988711988611988711988611988721198861198871198861198872119886119887119886
495
495
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886
4119861 11880 119887119886119887211988611988721198861198871198861198872119886119887119886
4119862 11880 1198872119886119887
5119860 19008 119887119886119887119886
5119861 47520 1198861198872119886119887119886119887119886
6119860
11880 119886119887119886
3960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198871198861198872119886
11988611988711988611988721198861198872119886
611986115840 119886119887
2119886119887119886119887
15840 1198861198872119886119887119886119887119886119887119886119887119886119887
611986215840 119887
21198861198872119886119887119886
15840 1198872119886119887119886119887119886119887119886119887119886119887
21198861198872
611986347520 119886119887
2119886119887119886119887119886119887119886
15840 1198861198872119886119887119886119887119886119887
2119886119887119886
Algebra 11
Table 35 Continued
119883119862
|O| 119892
6119864 15840 119886119887119886119887119886119887119886119887211988611988711988611988721198861198872119886119887
7119860 47520 1198871198861198872119886119887119886
811986147520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886
119886119887119886119887119886119887
911986095040 119887119886119887119886119887119886119887
31680 119887119886119887119886119887211988611988721198861198871198861198872119886
911986195040 119887
2119886119887119886119887119886
31680 1198861198872119886119887119886119887119886119887119886119887119886119887119886
10119860 95040 119886119887
1011986147520 119887119886119887119886119887
21198861198872119886
47520 11988711988611988721198861198872119886119887119886
11119860 190080 119886
12119860 47520 1198872119886119887119886
12119861
47520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198872119886119887119886119887119886119887
2
119886119887119886119887119886119887119886119887119886
15840
15840
1003816100381610038161003816100381610038161003816100381610038161003816
11988721198861198872119886119887119886119887
119887119886119887119886119887119886
1211986247520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198872119886119887119886
1198861198871198861198871198861198871198861198871198861198871198861198872
12119864
47520 11988721198861198872119886
47520 119886119887119886119887
47520 119887119886119887119886119887119886119887119886119887
47520 1198872119886119887119886119887119886119887119886
1411986047520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198871198861198872119886119887119886119887
119887
15119862 190080 119886119887119886119887119886
1811986095040 119887119886119887
2119886119887119886119887119886
95040 11988611988721198861198872119886119887119886119887
1811986195040 119887
21198861198872119886119887119886119887119886
95040 1198871198861198871198861198871198861198872119886
2011986095040
95040
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198872119886119887119886
119887119886
Table 36
119883119862
|O| 119892
2119860 630 119886119887211988621198872119886
3119861 8960 11988611988721198862
3119862 1920 11988621198871198862
Table 37
119883119862
|O| 119892
4119861 30240 1198861198872119886
5119861 48384 1198861198871198862
6119862 80640 119886
Table 38
119883119862
|O| 119892
2119860 495 119887119886119887119886211988711988621198871198861198872119886119887
2119861396 119886119887
311988611988721198862119887119886
396 11988621198873119886119887211988621198871198862
31198601320 119887119886
211988711988621198872119886119887
440 11988611988721198861198871198861198872119886119887119886119887
31198612640 119886
211988721198861198871198861198873119886
1584 119886119887119886119887119886211988731198862
3119862 7920 11988711988611988721198861198872119886119887
4119860 2970 1198871198861198871198861198871198861198872119886119887119886
4119861
11880 1198862119887
5940 1198872
2970 1198871198861198871198861198872119886119887
51198609504 119887
21198861198871198862
9504 119887119886211988731198862119887119886
5119861 47520 11988621198872119886
611986011880 119887119886
211988721198861198872119886
3960 119886119887211988621198871198861198871198862119887119886
611986115840 119886
2
15840 119886
6119862
23760 1198871198861198871198861198873119886
23760 11988711988621198872119886119887119886
15840 1198871198861198873119886119887119886
611986423760 119887
2119886
23760 11988621198872
711986095040 119887119886
2
31680 1198871198861198872119886119887119886
811986023760 119887119886119887119886119887
2119886
23760 1198872119886119887119886119887119886119887
811986123760 119886119887119886119887
2119886
23760 11988721198862119887119886
3142 119883 = 2119861 119905 = 119886 See Table 33
3143 119883 = 2119862 119905 = 119888 See Table 34
3144 119883 = 2119863 119905 = (119888(119888119889)12)11 See Table 35
315 119870 cong 1198621199003
3151 119883 = 2119860 119905 = 1198872 See Tables 36 and 37
12 Algebra
Table 39
119883119862
|O| 119892
8119862
23760 11988611988721198861198872
23760 1198872119886211988721198862
23760 1198872119886119887119886
23760 11988711988621198871198861198873
911986031680 119887119886119887
21198861198873
31680 119886119887119886119887119886119887119886119887
9119861 63360 1198861198871198861198871198861198872
1011986047520 119886119887119886
21198872
47520 1198871198861198871198861198872
1011986147520 119886119887119886
47520 119886211988731198862
1211986023760 119886
21198871198862
23760 1198861198873119886
12119861
47520 119887
47520 1198861198871198861198871198862
47520 1198871198861198872119886119887
1411986095040 119887119886119887119886
95040 1198861198872119886119887119886
1511986095040 119886119887
2
95040 119886119887119886119887
15119861 190080 1198861198871198861198872
1811986095040 119886119887
2119886119887
95040 119887119886119887119886119887
21119860 190080 11988711988621198871198862
2411986095040 119886119887
2119886
95040 119886211988721198862
2411986195040 119886
2119887119886
95040 119887119886
3011986095040 119886119887119886
2
95040 11988611988731198862
3152 119883 = 2119861 119905 = ((1198862119887)3119886119887)5 See Tables 38 and 39
316 119870 cong 1198621199002
3161 119883 = 2119860 119905 = 119886 See Table 40
3162 119883 = 2119861 119905 = 119886(119886119887119886119887119886)2119886(119886119887119886119887119886)7 See Table 41
3163 119883 = 2119862 119905 = (119887minus1(1198861198872)2119886119887minus1(1198861198872)4)5 See Table 42
Table 40
119883119862
|O| 119892
2119861 1260 1198871198861198872119886119887
2119862 1008 11988731198861198871198861198873
3119861 14336 119887119886119887
4119862 40320 119887
Table 41
119883119862
|O| 119892
2119860 15 (1198874119886)2119887311988611988721198861198874(119887119886119887)
2119886(119887119886119887)
711988611988711988611988731198861198872(119886119887)2
21198611680 119887
31198861198871198861198873
210 11988741198861198874119886119887119886119887
2119862 2520 (119886119887)5119886(119886119887)
2119886(119886119887)
3
3119861 35840 1198872119886119887
4119860 1920 11988711988611988731198861198872119886119887119886119887
4119862 20160 1198871198861198872119886119887
4119864 161280 119887119886119887
411986513440 119887119886119887119886119887119886119887119886119887
13440 119887311988611988711988611988721198861198872
5119861 344064 1198872
6119864 430080 119887
Table 42
119883119862
|O| 119892
2119860240 119886119887
4119886119887211988611988711988611988741198861198873119886119887311988611988741198861198873119886119887119886
45 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887411988611988741198861198872119886
2119861
720 1198872119886
240 11988741198861198874119886119887311988611988741198861198874119886119887211988611988731198861198872119886
180 119886119887211988611988731198861198872119886119887119886119887311988611988731198861198872
30 1198861198871198861198874119886
2119862
1440 1198861198871198861198873119886119887311988611988721198861198871198861198872119886
1440 119886119887311988611988741198861198873119886119887211988611988721198861198874119886
1440 1198872119886119887119886119887119886119887
311988611988741198861198873
1152 11988611988711988611988731198861198873119886119887311988611988741198861198872119886
360 11988611988711988611988711988611988731198861198872119886119887119886119887311988611988731198861198872
3119860 10240 1198861198874119886119887211988611988711988611988721198861198871198861198872119886
3119861
15360 11988611988731198861198872119886
7680 11988731198861198872
1024 1198861198872119886119887211988611988731198861198873119886
41198603840 119887
411988611988741198861198873119886119887411988611988741198861198874
1440 1198861198871198861198872119886119887211988611988741198861198872
4119861
11520 1198871198861198874119886119887411988611988731198861198873119886
5760 119886119887211988611988711988611988721198861198871198861198873119886
5760 11988611988721198861198874119886119887311988611988741198861198873119886
1440 11988611988741198861198871198861198872119886119887411988611988721198861198871198861198873119886
1440 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887119886
720 119886119887119886119887411988611988721198861198872119886119887211988611988731198861198873
720 1198872119886119887411988611988721198861198872119886119887411988611988731198861198874119886
Algebra 13
Table 42 Continued
119883119862
|O| 119892
4119862
11520 11988611988721198861198873119886
11520 1198861198873119886119887119886119887311988611988721198861198872
5760 119887311988611988741198861198872119886
5760 11988611988731198861198871198861198872
2880 11988611988711988611988741198861198872119886
1440 119887411988611988741198861198873119886119887211988611988721198861198873119886
1440 1198861198872119886119887311988611988731198861198872119886119887119886119887
720 11988611988711988611988711988611988711988611988721198861198871198861198873119886
720 119886119887311988611988741198861198872119886
4119863
23040 1198871198861198873119886119887119886119887211988611988731198861198872
11520 119886119887119886119887311988611988731198861198871198861198873
5760 119886119887411988611988711988611988741198861198872119886
2880 11988611988711988611988711988611988741198861198872119886119887411988611988741198861198873119886
2880 1198861198872119886119887119886119887119886119887
311988611988711988611988741198861198874119886
4119864
23040 1198871198861198871198861198871198861198872
23040 119886119887311988611988721198861198872
11520 1198861198873119886119887211988611988741198861198872
11520 119887311988611988711988611988731198861198872119886
5760 11988731198861198874119886
5760 1198861198871198861198872
5760 1198874119886119887311988611988721198861198871198861198873119886
4119865
23040 119886119887119886119887211988611988721198861198872
23040 1198873119886119887311988611988731198861198874119886
11520 119887411988611988741198861198872119886119887411988611988731198861198872119886
11520 119886119887311988611988741198861198873119886119887211988611988721198861198872
5760 1198872119886119887411988611988721198861198872119886119887211988611988731198861198872119886
5760 1198872119886119887211988611988731198861198873119886119887411988611988721198861198873119886
5760 1198861198873119886119887211988611988741198861198872119886119887311988611988731198861198872119886
5760 1198861198874119886119887119886119887119886119887
3119886119887211988611988721198861198873
5760 119886119887211988611988731198861198871198861198872119886119887211988611988731198861198873
5760 1198874119886119887411988611988721198861198872119886119887211988611988731198861198872119886
2880 119887311988611988721198861198871198861198872
2880 1198873119886119887411988611988731198861198872
1920 11988611988711988611988741198861198872119886119887211988611988741198861198873
1920 1198871198861198873119886119887311988611988711988611988721198861198874
4119866
46080 1198873119886119887311988611988721198861198873119886
46080 1198861198872119886119887311988611988721198861198872
23040 1198861198871198861198872119886119887211988611988731198861198872119886
23040 11988611988731198861198872119886119887311988611988731198861198874119886
5119860 73728 119886119887211988611988731198861198871198861198872119886
5119861184320 119887
21198861198872119886
122880 119887211988611988731198861198872
Table 42 Continued
119883119862
|O| 119892
6119860 92160 1198871198861198873119886119887411988611988731198861198872119886
6119861 122880 119887211988611988741198861198872119886
6119862
46080 1198872119886119887311988611988731198861198872119886
46080 119887311988611988711988611988721198861198872
15360 1198874119886119887211988611988731198861198873119886
15360 119886119887211988611988721198861198873119886119887
6119863
92160 119887411988611988711988611988721198861198874119886
23040 11988741198861198873119886119887411988611988721198861198872
15360 11988611988721198861198872119886119887119886
15360 119886119887411988611988731198861198873119886
6119864
92160 1198871198861198871198861198872
92160 1198871198861198874119886
46080 1198873119886119887211988611988731198861198872119886
46080 11988711988611988721198861198873119886
46080 11988721198861198871198861198874119886
46080 11988611988711988611988741198861198873
15360 119886119887211988611988711988611988741198861198871198861198872119886
6119865
92160 1198861198873119886119887119886
92160 11988611988741198861198872119886
46080 119887311988611988721198861198871198861198872119886
46080 1198861198874119886119887411988611988731198861198873
46080 1198861198873119886119887411988611988731198861198872
46080 11988721198861198872119886119887119886119887119886
7119860 737280 11988741198861198872
811986092160 119887
2119886119887119886
92160 11988611988741198861198873
8119861
92160 119886119887411988611988721198861198872119886119887119886
92160 119886119887411988611988731198861198873119886119887119886
46080 11988611988711988611988731198861198872119886
46080 119886119887311988611988721198861198874119886
23040 119887119886119887311988611988731198861198871198861198873
23040 11988721198861198874119886119887211988611988721198861198874
8119862
92160 1198873119886119887311988611988731198861198874
92160 1198872
92160 1198873
92160 119887119886119887211988611988721198861198872
8119863
92160 119887311988611988731198861198872
92160 11988611988721198861198873119886119887311988611988741198861198872
92160 119887411988611988731198861198873119886
92160 11988711988611988721198861198871198861198873119886
92160 119887311988611988721198861198872
14 Algebra
Table 42 Continued
119883119862
|O| 119892
8119864
92160 1198861198873119886
92160 1198861198872119886
46080 119886119887311988611988711988611988741198861198872
46080 119887311988611988711988611988741198861198872119886
46080 119886119887211988611988721198861198871198861198873
46080 1198872119886119887119886119887119886
46080 1198872119886119887411988611988731198861198873119886
46080 1198872119886119887211988611988731198861198873119886
46080 1198861198872119886119887211988611988731198861198873
46080 119886119887411988611988741198861198873
23040 1198861198874119886119887211988611988721198861198871198861198873
23040 11988741198861198874119886119887211988611988741198861198873
8119865
184320 11988611988741198861198871198861198873119886
184320 11988611988721198861198874119886119887119886
92160 119886119887119886119887311988611988711988611988721198861198872
92160 11988731198861198874119886119887411988611988731198861198872119886
9119860737280 119886119887
41198861198871198861198872
245760 119887211988611988741198861198872
10119860 368640 119887211988611988731198861198872119886
10119861368640 119886119887
21198861198872
368640 11988731198861198873119886
10119862368640 119886119887
4
368640 11988611988711988611988731198861198871198861198873119886
11119860 1474560 11988721198861198873119886
12119860122880 119887119886119887
21198861198871198861198874119886
122880 119887411988611988711988611988741198861198873
12119861368640 119887
211988611988711988611988731198861198872
184320 1198861198874119886119887211988611988741198861198872
12119862368640 119886119887
2119886119887211988611988731198861198872
184320 119886119887411988611988711988611988721198861198872
12119863
184320 119887411988611988711988611988721198861198872
184320 11988611988711988611988711988611988731198861198873119886
92160 119886119887411988611988721198861198873119886
92160 11988611988721198861198873119886119887119886
12119864368640 119887
311988611988721198861198874
368640 11988711988611988731198861198872
12119865
184320 11988721198861198871198861198872119886
184320 11988611988731198861198874119886
184320 11988611988711988611988731198861198873
184320 1198861198871198861198872119886
92160 1198861198872119886119887119886
92160 11988611988741198861198873119886
12119866 368640 11988711988611988741198861198872119886
Table 42 Continued
119883119862
|O| 119892
12119867
184320 1198874119886119887311988611988731198861198872
184320 119886119887311988611988721198861198873119886
184320 11988611988711988611988711988611988741198861198872119886
184320 1198861198874119886119887311988611988741198861198872119886
184320 119887311988611988721198861198872119886119887
184320 119886119887211988611988731198861198872119886
14119860 737280 1198861198874119886
15119860737280 119887
4119886
737280 1198871198861198872119886
16119860
368640 11988611988731198861198872
368640 11988611988721198861198871198861198874
368640 11988731198861198872119886
368640 11988711988611988741198861198873119886
18119860 737280 11988731198861198873
20119860737280 119886119887
41198861198872
737280 1198873119886119887119886
20119861737280 119886119887
41198861198874119886
737280 119886119887119886119887119886
24119860737280 119886119887
2
737280 1198873119886
24119861737280 119887
4
737280 119887
28119860737280 119886119887119886119887
31198861198872
737280 119886119887411988611988721198861198872
30119860737280 119887
21198861198874
737280 1198871198861198873
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R H Bing ldquoA homeomorphism between the 3-sphere andthe sum of two solid horned spheresrdquo Annals of MathematicsSecond Series vol 56 pp 354ndash362 1952
[2] D Montgomery and L Zippin ldquoExamples of transformationgroupsrdquo Proceedings of the American Mathematical Society vol5 pp 460ndash465 1954
[3] J W Morgan and H Bass Eds The Smith Conjecture vol 112of Pure and Applied Mathematics Academic Press OrlandoFla USA 1984 Papers presented at the symposium held atColumbia University New York NY USA 1979
[4] C M Edwards and G T Ruttimann ldquoInvolutive and Peircegradings in JBWlowast-triplesrdquo Communications in Algebra vol 31no 6 pp 2819ndash2848 2003
Algebra 15
[5] J Gornicki and B E Rhoades ldquoA general fixed point theoremfor involutionsrdquo Indian Journal of Pure and Applied Mathemat-ics vol 27 no 1 pp 13ndash23 1996
[6] S Montgomery Fixed Rings of Finite Automorphism Groupsof Associative Rings vol 818 of Lecture Notes in MathematicsSpringer Berlin Germany 1980
[7] T A Springer ldquoThe classification of involutions of simplealgebraic groupsrdquo Journal of the Faculty of Science University ofTokyo Section IA Mathematics vol 34 no 3 pp 655ndash670 1987
[8] T A Springer ldquoSome results on algebraic group s with invo-lutionsrdquo in Algebraic Groups and Related Topics (KyotoNagoya1983) vol 6 of Advanced Studies in Pure Mathematics pp 525ndash543 North-Holland Amsterdam The Netherlands 1985
[9] W Feit and J GThompson ldquoSolvability of groups of odd orderrdquoPacific Journal of Mathematics vol 13 pp 775ndash1029 1963
[10] R Brauer and K A Fowler ldquoOn groups of even orderrdquo Annalsof Mathematics vol 62 no 3 pp 565ndash583 1955
[11] D Gorenstein The Classification of Finite Ssimple Groups Vol1 Groups of Noncharacteristic 2 Type The University Series inMathematics Plenum Press New York NY USA 1983
[12] D Gorenstein R Lyons and R Solomon The Classification ofthe Finite Simple Groups vol 40 of Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 1994
[13] C Bates and P Rowley ldquoInvolutions in Conwayrsquos largest simplegrouprdquo LMS Journal of Computation andMathematics vol 7 pp337ndash351 2004
[14] P Rowley and P Taylor ldquoPoint-line collinearity graphs of twosporadic minimal parabolic geometriesrdquo Journal of Algebra vol331 pp 304ndash310 2011
[15] P Rowley and P Taylor ldquoInvolutions in Jankorsquos simple group 1198694rdquoLMS Journal of Computation and Mathematics vol 14 pp 238ndash253 2011
[16] P Taylor ldquoInvolutions in Fischerrsquos sporadic groupsrdquo preprinthttpeprintsmamanacuk1622
[17] The GAP Group GAP-Groups Algorithms Programming Ver-sion 43 2002 httpwwwgap-systemorg
[18] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I The user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997
[19] R Wilson P Walsh J Tripp et al ldquoAtlas o f finite grouprepresentationsrdquo httpbrauermathsqmulacukAtlasv3
[20] J H Conway R T Curtis S P Norton R A Parker and R AWilson Atlas of Finite Groups Clarendon Oxford UK 1985
[21] R AWilson ldquoStandard generators for sporadic simple groupsrdquoJournal of Algebra vol 184 no 2 pp 505ndash515 1996
[22] D Gorenstein Finite Groups Chelsea Publishing New YorkNY USA 2nd edition 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Algebra 5
Table 11
119883119862
|O| 119892
2119860 15 119887(119887119886)2119887(119887119886)
2119887(119887119886)
2
2119861
15
15
1003816100381610038161003816100381610038161003816100381610038161003816
((119887119886)2119887)3(119887119886)3
119887(119887119886)2(119887(119887119886)
31198872119886)2
1
1
1003816100381610038161003816100381610038161003816100381610038161003816
(119887119886)21198872119886(119887(119887119886)
2)21198872119886119887(119887119886)
3
1198872119886119887(119887119886)
3119887(119887119886)
2119887(119887119886)
3(1198872119886)2
3119860 24 119887(119887119886)2119887(119887119886)
2
3119861 60 119887(119887119886)3119887(119887119886)
2119887(119887119886)
3119887(119887119886)
4119860 60 (119887119886)119887(119887119886)2
5119860 60 (119887119886)2119887(119887119886)
2119887(119887119886)
3119887(119887119886)119887(119887119886)
2
5119861 60 (119887119886)119887(119887119886)119887(119887119886)3119887(119887119886)
2119887(119887119886)
3
5119862120 119887(119887119886)
3119887(119887119886)
3
24 119887(119887119886)119887(119887119886)2119887(119887119886)
3
5119863120 (119887119886)119887(119887119886)
2119887(119887119886)
3
24 (119887119886)2119887(119887119886)
2119887(119887119886)119887(119887119886)
6119860 120 (119887119886)3
611986160
60
1003816100381610038161003816100381610038161003816100381610038161003816
(119887119886)2119887(119887119886)
3
(119887119886)3119887(119887119886)
4
7119860 240 119887(119887119886)4
1011986060 119887(119887119886)119887(119887119886)
2119887(119887119886)
2
60 119887(119887119886)2119887(119887119886)
3
1011986160 (119887119886)
2119887(119887119886)
2119887(119887119886)
60 (119887119886)119887(119887119886)2119887(119887119886)119887(119887119886)
10119862120 (119887119886)119887(119887119886)119887(119887119886)
120 (119887119886)2119887(119887119886)
10119863120 119887(119887119886)
3
120 119887(119887119886)119887(119887119886)2
12119860120
120
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119887119886)
(119887119886)
15119860 240 (119887119886)119887(119887119886)119887(119887119886)2119887(119887119886)
2
15119861 240 (119887119886)119887(119887119886)2119887(119887119886)119887(119887119886)
2
Table 12
119883119862
|O| 119892
2119860 21 (1198871198861198872119886)2(119887119886)3
2119861 28 1198871198861198872
311986014
14
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)3(1198861198872)2119886119887
(1198872119886119887119886)2119887119886119887
3119861 168 119887119886119887119886
411986084 119886119887119886119887
2119886119887
42 1198872119886119887119886119887
6119860
84
84
1003816100381610038161003816100381610038161003816100381610038161003816
1198872
119887
42
42
1003816100381610038161003816100381610038161003816100381610038161003816
119886119887119886119887119886119887119886
1198861198871198861198872119886119887119886119887
6119861 168 119886119887
7119860 336 119887119886119887
8119860168 119886119887
2119886
168 119886119887119886
12119860168
168
1003816100381610038161003816100381610038161003816100381610038161003816
119887119886
119886
Table 13
119883119862
|O| 119892
211986084 119887119886119887
3119886119887
14 11988731198861198872119886119887
3119860 448 1198872
4119860
336 119887119886119887119886119887
336 119887119886119887119886119887119886119887
336 1198871198861198871198861198871198861198872
5119860 896 119887119886119887
6119860 1344 119887
Table 14
119883119862
|O| 119892
211986080 119887
211988611988731198861198872
30 119887211988611988711988611988721198861198871198861198872
3119860 640 119887119886119887
4119861 480 1198871198861198873119886119887
4119862 960 1198872119886119887
5119861 128 1198871198861198872119886119887
5119862 1536 119887
6119861 1920 1198872
Table 15
119883119862
|O| 119892
211986045 119887119886119887
3119886119887
30 11988741198861198874119886119887119886119887
211986136
36
1003816100381610038161003816100381610038161003816100381610038161003816
119887211988611988731198861198872
1198871198861198871198861198871198861198873119886
3119860360 119887
31198861198872119886119887119886119887
24 1198861198872119886119887211988611988711988611988721198861198872119886
4119861 180 119886
4119862
360 11988711988611988711988611988741198861198872119886
180
180
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198871198861198872
1198872119886119887119886119887119886119887
5119860144
144
1003816100381610038161003816100381610038161003816100381610038161003816
11988711988611988741198861198873119886119887
11988711988611988731198861198874119886119887
5119861 720 1198871198861198871198861198872119886
Table 16
119883119862
|O| 119892
6119861
720 119887119886119887
360 1198871198861198872119886
360 1198861198872119886119887
7119860 2880 119887
8119861720 119887
41198861198872119886
720 1198861198873119886119887
8119862720 119887119886119887
3119886
720 11988611988721198861198874
10119860720
720
1003816100381610038161003816100381610038161003816100381610038161003816
1198872
1198873
10119861720
720
1003816100381610038161003816100381610038161003816100381610038161003816
119886119887119886
119887119886
15119860 2880 1198872119886
6 Algebra
Table 17
119883119862
|O| 119892
2119860 105 119886
2119861 28 119887119886
3119860 336 1198872
4119861 630 119887
Table 18
119883119862
|O| 119892
211986060 119886119887
31198861198871198861198872119886
5 11988611988731198861198872119886
2119861 60 1198872119886
3119860240 119886
160 11988611988731198861198871198861198872119886119887119886
4119860 120 1198871198861198873119886119887119886
4119861
240 119886119887
120 1198871198861198872119886119887119886119887
30 1198861198872119886
4119862120 119886119887
21198861198872119886
120 1198861198873119886119887119886
5119860192 119886119887119886119887
3
192 11988711988611988721198861198873
5119861 320 1198874119886119887
5119862 1920 11988721198861198872
6119860480 119886119887
2119886119887
480 1198871198861198872119886
Table 19
119883119862
|O| 119892
6119861480 119887
4119886119887119886119887
480 11988721198861198872119886
7119860 1920 119887
8119860480 119887119886119887119886119887
2
480 11988711988611988721198861198872119886119887
10119860960 119887
211988611988721198861198872119886
960 119887119886119887
10119861 960 1198861198872
11119860 1920 1198872119886119887119886
11119861 1920 119887119886119887119886119887
12119860960 119887119886
960 1198873
15119860 1920 1198871198861198873
20119860960 119887119886119887119886
960 1198872119886119887
20119861960 119887
21198861198873119886
960 1198874119886119887119886
Table 20
119883119862
|O| 119892
2119860120 119887(119887119886)
3119887(119887119886)119887(119887119886)
5
10 (119887119886)119887(119887119886)4119887(119887119886)
2
3119860 192 119887(119887119886)6
3119861 640 (119887119886)119887(119887119886)2
4119860480 119887(119887119886)
5119887(119887119886)
80 (119887119886)3119887(119887119886)
3119887(119887119886)
4
5119860960 (119887119886)
3119887(119887119886)
3119887(119887119886)
2
320 (119887119886)2119887(119887119886)
2
5119861960 (119887119886)119887(119887119886)
3119887(119887119886)
4
320 (119887119886)119887(119887119886)3
6119860 960 (119887119886)119887(119887119886)6
8119860960 (119887119886)
4119887(119887119886)
2
960 (119887119886)119887(119887119886)5
9119860 1920 119887(119887119886)2119887(119887119886)
2
9119861 1920 (119887119886)
9119862 1920 119887(119887119886)2
10119860960 (119887119886)
4119887(119887119886)
960 119887(119887119886)4119887(119887119886)119887(119887119886)
10119861960 119887(119887119886)119887(119887119886)119887(119887119886)119887(119887119886)
2
960 119887(119887119886)5
12119860960
960
1003816100381610038161003816100381610038161003816100381610038161003816
(119887119886)2119887(119887119886)
4
(119887119886)6
15119860 1920 (119887119886)2119887(119887119886)
15119861 1920 119887(119887119886)3
17119860 1920 (119887119886)2
17119861 1920 (119887119886)2119887(119887119886)
3
Table 21
119883119862
|O| 119892
2119860 153 1198861198872119886119887119886119887119886119887
3119860102
102
1003816100381610038161003816100381610038161003816100381610038161003816
1198872
119887
3119861 272 1198861198871198861198872119886
4119860612 119887119886119887119886119887119886119887
2119886
306 1198871198861198872119886119887119886119887119886
6119860
612
612
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198872119886119887119886
1198871198861198871198861198872119886
306
306
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198872119886119887119886119887119886
119887119886119887119886
81198601224 119886119887119886119887119886119887
1224 1198872119886119887119886119887119886
9119860 816 11988721198861198871198861198872119886119887
9119861 816 1198871198861198871198861198872119886119887
9119862 816 1198871198861198872
121198601224
1224
1003816100381610038161003816100381610038161003816100381610038161003816
119886119887
1198861198872119886119887119886
17119860 2448 119887119886
17119861 2448 119886
19119860 2448 1198861198872119886119887
19119861 2448 1198872119886119887119886
Algebra 7
Table 22
119883119862
|O| 119892
2119860
168 1198872119886119887119886119887119886119887
2119886119887
14 119886119887119886
14 1198861198871198861198872119886119887
2119861 84 1198872119886119887
3119860 896 119887
3119861 128 119887211988611988721198861198871198861198872119886119887119886119887
Table 23
119883119862
|O| 119892
4119860112 119887119886119887119886119887119886119887
112 1198872119886119887211988611988721198861198872
4119861
1344 1198861198872119886119887119886
672 11988611988721198861198871198861198872119886119887
672 11988721198861198871198861198872119886119887119886
5119860 1792 119887119886119887119886
6119860 5376 119886
Table 24
119883119862
|O| 119892
211986060 119887
2119886119887119886119887211988611988721198861198871198861198872119886119887119886119887119886119887
2119886119887
15 11988711988611988721198861198872119886119887119886119887119886119887
2
2119861
120 119887119886119887211988611988711988611988721198861198871198861198872
80 119887211988611988711988611988721198861198871198861198872119886119887
2 (119887119886)8(1198872119886)21198871198861198872(119886119887)2(1198861198872)2(119886119887)3(1198861198872119886119887)2119887
3119860 320 119887211988611988721198861198871198861198872119886119887119886119887119886119887
3119861 960 11988721198861198871198861198872119886119887
4119860120 119887119886119887
21198861198872
120 1198871198861198871198861198872
4119861
480 1198872119886119887119886119887119886119887
2119886119887
240 1198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
240 1198872119886119887119886119887119886119887119886119887119886119887119886119887
4119862
960 1198871198861198872119886119887
960 11988721198861198871198861198872
160 1198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
21198861198872
160 1198872119886119887119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
5119860 1920 119887119886119887
6119860 1920 1198871198861198871198861198871198861198872
61198611920 119887119886119887
2
1920 1198872119886119887
101198601920 119887
21198861198871198861198872119886119887119886119887
1920 1198872119886119887119886119887119886119887119886119887
2
11119860 7680 119887119886119887119886119887
121198613840 119887
3840 1198872
3102 119883 = 2119861 119905 = 119886 See Table 24
311 119870 cong 119872119888119871
3111119883 = 2119860 119905 = 119886 See Table 25
3112119883 = 2119861 119905 = 119888 See Table 26
Table 25
119883119862
|O| 119892
2119860 210 119887211988611988731198861198872
3119861 2240 1198871198861198871198861198872
4119860 5040 1198872119886119887
5119861 8064 119887
6119861 6720 1198872
Table 26
119883119862
|O| 119892
2119860 165 11988611988721198861198871198861198872119886119887
3119860 220 1198872119886119887311988611988721198861198871198861198872119886
3119861660 119886119887
21198861198872
220 1198872119886119887119886119887119886119887
3
41198601980 119887
21198861198873119886119887119886119887119886
990 119886
5119860792 119887
211988611988721198861198873
792 1198872119886119887119886119887119886119887
2
5119861 7920 1198872
6119860 1980 119886119887
6119861
1980 1198871198861198872119886119887119886119887
1980 1198861198872119886119887
660
660
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198873
11988711988611988731198861198874119886119887
71198603960 119887119886119887119886119887
2119886
3960 1198871198861198872
71198613960 119887119886119887
3119886
3960 1198872119886
81198603960 119887
21198861198872119886
3960 11988611988731198861198873
9119860 2640 1198872119886119887119886
9119861 2640 1198861198871198861198872119886119887
101198603960 119887119886119887119886119887
3960 119886119887119886119887119886
121198603960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198873
1198871198861198873
141198603960 119886119887
21198861198873
3960 1198871198861198872119886119887
141198613960 119887119886119887119886119887119886
3960 1198872119886119887119886119887119886
151198603960 119887119886119887
3960 1198874
151198613960 119886119887119886119887
3960 119887
301198603960 119887119886
3960 1198871198861198872119886
301198613960 119887
41198861198872119886
3960 1198861198872
8 Algebra
Table 27
119883119862
|O| 119892
2119860240 119887119886119887119886119887119886119887
2 1198875(1198861198876)2(119886119887)31198875119886(11988721198861198873)2
2119861 315 119887211988611988711988611988721198861198872119886119887
3119860 1344 1198872119886119887
3119861 960 1198871198861198874119886119887
4119860480 119887
31198861198871198861198872119886119887
480 11988721198861198871198861198873119886119887
41198612520
2520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887
1198874119886119887
5119860 2688 1198873
6119860 13440 119887
Table 28
119883119862
|O| 119892
2119860 84 (119886119887)21198875(119886119887)2
2119861
168
168
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)2
(119886119887)1198874(119886119887)119887
2(119886119887)119887
2(119886119887)119887
4
14
14
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198874(119886119887)119887
4(119886119887)119887
2(119886119887)119887
2
1198872(119886119887)119887
2(119886119887)21198874(119886119887)119887
2(119886119887)119887
2
3119860 896 1198874(119886119887)119887
2
3119861 2688 1198872
4119861
1344
1344
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)119887
2(119886119887)119887
2
1198872(119886119887)119887
4(119886119887)21198874
112
112
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)1198872(119886119887)41198874(119886119887)2
(119886119887)31198874(119886119887)41198872
4119862
1344
1344
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)1198874(119886119887)119887
4
1198874(119886119887)119887
2(119886119887)119887
2(119886119887)31198872
672
672
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198872(119886119887)3
1198872(119886119887)41198872(119886119887)2
672
672
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)41198872(119886119887)2
(119886119887)4119887(119886119887)
2
5119860 5376 (119886119887)3
6119860 5376 (119886119887)1198872(119886119887)119887
2(119886119887)4
61198615376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198872(119886119887)
(119886119887)1198872(119886119887)
7119862 3072 1198872(119886119887)119887
2(119886119887)119887
2(119886119887)2
81198605376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)21198872(119886119887)21198872
(119886119887)1198872(119886119887)21198872(119886119887)
101198605376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)2
1198872(119886119887)119887
2(119886119887)119887
2
1211986110752
10752
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)2
1198874(119886119887)2
15119860 21504 1198872(119886119887)3
17119860 21504 (119886119887)1198872(119886119887)2
17119861 21504 (119886119887)
21119860 21504 (119886119887)1198872(119886119887)119887
2
21119861 21504 1198872(119886119887)
Table 29
119883119862
|O| 119892
2119860105 (119886119887)119887(119886119887)
3119887(119886119887)
3
63 119887611988611988731198861198873119886119887211988611988761198861198872119886119887611988611988761198861198874119886
2119861 315 (1198861198872)2(119887119886)21198872119886
3119860630 (119886119887)119887(119886119887)119887(119886119887)119887
126 119887(119886119887)21198876
3119861 2520 (119886119887)1198876
41198601260 (119886119887)
21198875(119886119887)
1260 (119886119887)119887(119886119887)1198873
4119861
945
945
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)21198872(119886119887)119887
119887(119886119887)119887(119886119887)1198872(119886119887)119887
315
315
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119886119887)5119887
119887(119886119887)31198872(119886119887)119887
41198621890
1890
1003816100381610038161003816100381610038161003816100381610038161003816
1198873(119886119887)2
(119886119887)1198872(119886119887)2119887
5119860 2520 (119886119887)3119887(119886119887)
6119860
3780 (119886119887)31198872(119886119887)2
1890 119887(119886119887)119887(119886119887)
1890 119887(119886119887)119887(119886119887)119887(119886119887)3
1260 119887(119886119887)1198872(119886119887)119887(119886119887)
2
6119861 7560 (119886119887)
71198601890 (119886119887)119887(119886119887)
41198872
90 119887(119886119887)1198873(119886119887)2119887
71198611890 119887
5(119886119887)119887(119886119887)
90 1198875(119886119887)119887
5(119886119887)119887
2(119886119887)119887
5
7119862 2160 119887(119886119887)119887(119886119887)4
71198637560 (119886119887)119887(119886119887)
4
1080 (119886119887)1198873(119886119887)
71198647560 (119886119887)
3119887(119886119887)
2
1080 (119886119887)1198874(119886119887)2119887(119886119887)
8119860
3780
3780
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)2119887(119886119887)
2
(119886119887)4
3780
3780
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)3119887(119886119887)119887(119886119887)119887
(119886119887)119887(119886119887)119887(119886119887)2
10119860 7560 (119886119887)3
121198607560 119887(119886119887)119887
7560 119887(119886119887)1198872
121198617560
7560
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119886119887)2119887
119887
14119860
3780 (119886119887)1198873(119886119887)2
3780 (119886119887)21198872(119886119887)
1890 (119886119887)1198872(119886119887)
1890 119887(119886119887)3119887
14119861
3780 (119886119887)119887(119886119887)2119887(119886119887)
3780 (119886119887)1198874
1890 (119886119887)21198872(119886119887)119887(119886119887)
2
1890 119887(119886119887)119887(119886119887)2
Algebra 9
Table 29 Continued
119883119862
|O| 119892
141198627560 119887(119886119887)
7560 119887(119886119887)3
141198637560 119887
3(119886119887)
7560 119887(119886119887)2
15119860 15120 (119886119887)1198872
21119860 15120 (119886119887)4119887(119886119887)
21119861 15120 (119886119887)119887
21119862 15120 (119886119887)3119887
21119863 15120 1198872
281198607560 119887
2(119886119887)119887
2
7560 (119886119887)2
281198617560 119887
4(119886119887)3
7560 (119886119887)119887(119886119887)2
Table 30
119883119862|O| 119892
2119860
640 11988611988731198861198872119886119887119886
480 11988611988721198861198872119886119887119886119887119886119887119886119887119886119887
21198861198872119886
240 11988611988721198861198873119886119887311988611988731198861198873119886119887211988611988711988611988721198861198873119886119887119886119887119886119887119886119887
31198861198873
30 1198861198872119886119887119886119887119886119887211988611988721198861198873119886119887311988611988711988611988711988611988721198861198873119886119887211988611988721198861198873119886119887211988611988711988611988711988611988731198863119860 10240 119886119887
3119886119887119886119887119886119887119886119887119886119887
2119886
4119860 5120 119886119887119886119887119886119887119886
4119862
7680 119886119887119886119887211988611988721198861198872119886119887119886
3840 119886119887211988611988731198861198871198861198872119886119887119886
3840 119886119887119886119887211988611988711988611988731198861198872119886
411986315360 119886119887119886
7680 1198861198872119886119887211988611988711988611988721198861198871198861198872119886
5119860 12288 11988611988731198861198871198861198872119886119887119886119887119886119887119886
5119861 4096 119886119887119886119887119886
611986061440 119886119887
21198861198871198861198872119886119887119886
30720 119886119887119886119887119886119887119886119887119886119887119886
7119860 61440 119886
811986261440 119886119887119886119887
21198861198872119886
61440 11988611988721198861198872119886119887119886
10119860 61440 11988611988721198861198872119886119887119886119887119886119887119886
12119860122880 1198861198872119886119887119886
13119860 61440 1198861198873119886119887119886
312 119870 cong 119867119890
3121119883 = 2119860 119905 = 119886 See Table 27
3122119883 = 2119861 119905 = (1198861198872)6 See Table 28
3123 119883 = 2119862 119905 = 1198893 See Table 29
313 119870 cong 119877119906
3131 119883 = 2119860 119905 = 1198872 See Table 30
3132 119883 = 2119861 119905 = 119886 See Table 31
314 119870 cong 119878119906119911
3141119883 = 2119860 119905 = ((119886119887)21198861198872)6 See Table 32
Table 31
119883119862 |O| 119892
2119860 455 119887119886119887311988611988721198861198873119886119887
2119861
455 11988711988611988731198861198872119886119887119886119887119886119887119886119887119886119887
455 11988711988611988711988611988711988611988711988611988711988611988721198861198873119886119887
1 1198873119886119887311988611988731198861198872119886119887119886119887119886119887
211988611988721198861198873119886119887119886119887211988611988721198861198872
1 1198871198861198872119886119887119886119887311988611988711988611988721198861198872119886119887119886119887211988611988711988611988731198861198873119886119887119886119887119886119887
31198861198871198861198872119886119887119886119887211988611988731198861198871198861198873
3119860 5824 1198872119886119887119886119887119886119887119886119887119886119887
4119860 3640 (119886119887)10119886(119886119887)15119886((119886119887)
5119886)2
4119862 3640 1198873119886119887119886119887119886119887
21198861198872
4119863 3640 119887311988611988711988611988721198861198873119886119887
51198605824 119887
311988611988711988611988721198861198871198861198872
5824 119887211988611988731198861198872119886119887
5119861 29120 1198872119886119887119886119887
6119860 29120 1198872119886119887119886119887119886119887
2119886119887
7119860
29120 11988721198861198872
29120 119887
29120 1198872
1011986029120 119887119886119887119886119887
2119886119887119886119887
29120 119887119886119887119886119887
1011986129120 119887119886119887119886119887
31198861198873119886119887
29120 11988711988611988731198861198873119886119887119886119887
12119860 58240 1198873119886119887119886119887119886119887
2119886119887
13119860
29120 119887119886119887
29120 119887119886119887119886119887119886119887
29120 11988731198861198872119886119887
1411986029120 119887
311988611988721198861198872119886119887119886119887
29120 11988711988611988721198861198872119886119887119886119887119886119887
1411986129120 119887
21198861198872119886119887119886119887119886119887
29120 11988711988611988711988611988711988611988721198861198872
1411986229120 119887119886119887
2119886119887119886119887119886119887119886119887
2
29120 1198872119886119887119886119887119886119887119886119887
2119886119887
15119860 116480 11988731198861198872119886119887119886119887119886119887
2011986058240 119887
31198861198871198861198872
58240 11988721198861198873119886119887
2611986029120 119887119886119887
2
29120 1198872119886119887
2611986129120 119887119886119887
21198861198872
29120 11988721198861198872119886119887
2611986229120 119887
3119886119887119886119887119886119887
29120 11988711988611988711988611988721198861198872
29119860 116480 1198871198861198872119886119887
29119861 116480 1198873119886119887119886119887
Table 32
119883119862
|O| 119892
2119860360 119886
54 ((119886119887)2119886)4119886((119886119887)
2119886)3
3119861 5120 11988721198861198872119886119887119886119887119886119887119886119887119886119887119886
3119862 9216 119887119886119887119886
4119860 1728 119887211988611988721198861198872119886119887119886119887119886119887
4119862 17280 1198872119886119887
5119861 55296 1198872119886119887119886119887119886
6119863 46080 119887
10 Algebra
Table 33119883119862|O| 119892
2119860 315 1198871198861198872119886119887119886119887119886119887
21198861198872119886119887211988611988721198861198872119886119887119886119887119886119887
2119886119887
2119861
630 1198872119886119887119886119887119886119887119886119887
2119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
21198861198872119886119887119886119887211988611988721198861198872119886119887
560 119887119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
2 (1198871198861198872119886)2(1198872119886)2(119887119886(1198872119886)2)2(1198872119886119887119886)2((1198872119886)21198872119886119887119886)21198871198861198872(119886119887)2
3119860 224 (1198871198861198872119886)3119887(119887119886)3(1198872119886)2(119887119886)5119887(119887119886)4(1198872119886)2119887
3119861 2240 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887
3119862 20160 11988711988611988721198861198871198861198872119886119887
41198602520 119887119886119887119886119887119886119887
2119886119887211988611988721198861198872119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886119887119886119887
4119861
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
2520 11988711988611988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
21198861198872119886119887
1260 119887211988611988721198861198871198861198872119886119887119886119887
4119862
5040 11988711988611988721198861198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
2520 11988711988611988711988611988711988611988721198861198872119886119887119886119887119886119887
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
4119863
10080 11988711988611988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
10080 11988711988611988711988611988721198861198872119886119887119886119887211988611988721198861198872119886119887
1120 (119886119887)4((119886119887)4119886)4119886(119886119887)8((119886119887)4119886)2
1120 (119886(1198872119886)4)2(1198872119886)8119886(119886(1198872119886)4)4(1198872119886)4
5119860 20160 119887119886119887119886119887
5119861 40320 1198872119886119887119886119887119886119887
2119886119887
6119860 10080 1198871198861198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
6119863 20160 11988711988611988721198861198872119886119887
611986440320 119887
2119886119887119886119887119886119887
2119886119887119886119887119886119887
40320 1198872119886119887119886119887211988611988721198861198872119886119887119886119887119886119887
7119860 80640 1198872119886119887119886119887119886119887119886119887
811986020160 119887119886119887
2119886119887119886119887211988611988721198861198872119886119887
20160 1198871198861198872119886119887211988611988721198861198871198861198872119886119887
8119861
20160 1198871198861198872119886119887119886119887119886119887119886119887
211988611988721198861198871198861198872119886119887
20160 11988711988611988711988611988721198861198872119886119887
20160 11988711988611988721198861198872119886119887119886119887
20160 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887119886119887
2119886119887
10080 11988721198861198872119886119887
10080 1198872119886119887119886119887
8119862
20160 11988721198861198871198861198872119886119887119886119887119886119887
20160 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
2119886119887
20160 1198872119886119887211988611988721198861198871198861198872119886119887
20160 1198871198861198872119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
10119860
40320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
119887211988611988711988611988721198861198871198861198872119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886119887
20160 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887
20160 11988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
1011986140320 119887119886119887119886119887119886119887119886119887
2119886119887
40320 1198871198861198872119886119887119886119887119886119887119886119887
11119860 161280 11988721198861198871198861198872119886119887119886119887
1211986040320 119887
2119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
20160 119887211988611988711988611988721198861198872119886119887119886119887119886119887
2119886119887
1211986140320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886119887
11988721198861198872119886119887211988611988721198861198872119886119887
12119862 40320 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
1211986380640 119887119886119887
2119886119887119886119887119886119887
80640 1198871198861198871198861198871198861198872119886119887
13119860 161280 119887119886119887
13119861 161280 11988721198861198872119886119887119886119887
14119860 80640 119887211988611988711988611988721198861198872119886119887119886119887
15119860 161280 119887
15119861 161280 1198872119886119887
15119862 161280 119887211988611988721198861198871198861198872119886119887119886119887119886119887
Table 33 Continued119883119862
|O| 119892
2011986080640
80640
1003816100381610038161003816100381610038161003816100381610038161003816
119887119886119887211988611988721198861198872119886119887119886119887
119887119886119887119886119887211988611988721198861198872119886119887
21119860 161280 119887119886119887211988611988721198861198872119886119887119886119887119886119887
21119861 161280 119887119886119887211988611988721198861198872119886119887
2411986080640 119887119886119887119886119887
2119886119887
80640 1198871198861198872119886119887119886119887
Table 34
119883119862
|O| 119892
2119860 315 ((119886119887)3119886)2119886((119886119887)
3119886)4119886(119886119887)
3((119886119887)3119886)2119886
2119861 1800 1198861198871198861198871198861198872
3119860 200 119887211988611988721198861198871198861198872119886119887119886119887
3119862 16800 11988611988721198861198872119886119887119886119887119886
41198603150 119887119886119887
21198861198871198861198872119886
630 1198871198861198872119886119887119886119887119886119887119886119887119886119887
2119886
4119862 12600 1198871198861198872119886119887119886
5119861 20160 119886119887119886
6119860 12600 1198861198871198861198871198861198872119886119887
6119864 100800 119887119886
7119860 100800 119887
811986025200 119886119887119886119887119886119887
25200 119887119886119887119886119887119886119887
12119860 50400 119886
Table 35
119883119862
|O| 119892
2119860 495 (119886119887)5119886 (119886119887)
2(119886 (119886119887)
5)4
(119886119887)5
2119861 792 11988721198861198871198861198872119886119887119886119887119886119887
2119886119887119886119887
3119860 264 119886(119886119887)9((119886119887)9119886)4119886(119886119887)
9((119886119887)9119886)2
31198615280 119886(119886119887)
6((119886119887)3119886)4
1760 (119886((119886119887)3119886)2)4119886
3119862 2640 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886
4119860
2970 11988711988611988711988611988721198861198871198861198872119886119887119886
495
495
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886
4119861 11880 119887119886119887211988611988721198861198871198861198872119886119887119886
4119862 11880 1198872119886119887
5119860 19008 119887119886119887119886
5119861 47520 1198861198872119886119887119886119887119886
6119860
11880 119886119887119886
3960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198871198861198872119886
11988611988711988611988721198861198872119886
611986115840 119886119887
2119886119887119886119887
15840 1198861198872119886119887119886119887119886119887119886119887119886119887
611986215840 119887
21198861198872119886119887119886
15840 1198872119886119887119886119887119886119887119886119887119886119887
21198861198872
611986347520 119886119887
2119886119887119886119887119886119887119886
15840 1198861198872119886119887119886119887119886119887
2119886119887119886
Algebra 11
Table 35 Continued
119883119862
|O| 119892
6119864 15840 119886119887119886119887119886119887119886119887211988611988711988611988721198861198872119886119887
7119860 47520 1198871198861198872119886119887119886
811986147520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886
119886119887119886119887119886119887
911986095040 119887119886119887119886119887119886119887
31680 119887119886119887119886119887211988611988721198861198871198861198872119886
911986195040 119887
2119886119887119886119887119886
31680 1198861198872119886119887119886119887119886119887119886119887119886119887119886
10119860 95040 119886119887
1011986147520 119887119886119887119886119887
21198861198872119886
47520 11988711988611988721198861198872119886119887119886
11119860 190080 119886
12119860 47520 1198872119886119887119886
12119861
47520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198872119886119887119886119887119886119887
2
119886119887119886119887119886119887119886119887119886
15840
15840
1003816100381610038161003816100381610038161003816100381610038161003816
11988721198861198872119886119887119886119887
119887119886119887119886119887119886
1211986247520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198872119886119887119886
1198861198871198861198871198861198871198861198871198861198871198861198872
12119864
47520 11988721198861198872119886
47520 119886119887119886119887
47520 119887119886119887119886119887119886119887119886119887
47520 1198872119886119887119886119887119886119887119886
1411986047520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198871198861198872119886119887119886119887
119887
15119862 190080 119886119887119886119887119886
1811986095040 119887119886119887
2119886119887119886119887119886
95040 11988611988721198861198872119886119887119886119887
1811986195040 119887
21198861198872119886119887119886119887119886
95040 1198871198861198871198861198871198861198872119886
2011986095040
95040
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198872119886119887119886
119887119886
Table 36
119883119862
|O| 119892
2119860 630 119886119887211988621198872119886
3119861 8960 11988611988721198862
3119862 1920 11988621198871198862
Table 37
119883119862
|O| 119892
4119861 30240 1198861198872119886
5119861 48384 1198861198871198862
6119862 80640 119886
Table 38
119883119862
|O| 119892
2119860 495 119887119886119887119886211988711988621198871198861198872119886119887
2119861396 119886119887
311988611988721198862119887119886
396 11988621198873119886119887211988621198871198862
31198601320 119887119886
211988711988621198872119886119887
440 11988611988721198861198871198861198872119886119887119886119887
31198612640 119886
211988721198861198871198861198873119886
1584 119886119887119886119887119886211988731198862
3119862 7920 11988711988611988721198861198872119886119887
4119860 2970 1198871198861198871198861198871198861198872119886119887119886
4119861
11880 1198862119887
5940 1198872
2970 1198871198861198871198861198872119886119887
51198609504 119887
21198861198871198862
9504 119887119886211988731198862119887119886
5119861 47520 11988621198872119886
611986011880 119887119886
211988721198861198872119886
3960 119886119887211988621198871198861198871198862119887119886
611986115840 119886
2
15840 119886
6119862
23760 1198871198861198871198861198873119886
23760 11988711988621198872119886119887119886
15840 1198871198861198873119886119887119886
611986423760 119887
2119886
23760 11988621198872
711986095040 119887119886
2
31680 1198871198861198872119886119887119886
811986023760 119887119886119887119886119887
2119886
23760 1198872119886119887119886119887119886119887
811986123760 119886119887119886119887
2119886
23760 11988721198862119887119886
3142 119883 = 2119861 119905 = 119886 See Table 33
3143 119883 = 2119862 119905 = 119888 See Table 34
3144 119883 = 2119863 119905 = (119888(119888119889)12)11 See Table 35
315 119870 cong 1198621199003
3151 119883 = 2119860 119905 = 1198872 See Tables 36 and 37
12 Algebra
Table 39
119883119862
|O| 119892
8119862
23760 11988611988721198861198872
23760 1198872119886211988721198862
23760 1198872119886119887119886
23760 11988711988621198871198861198873
911986031680 119887119886119887
21198861198873
31680 119886119887119886119887119886119887119886119887
9119861 63360 1198861198871198861198871198861198872
1011986047520 119886119887119886
21198872
47520 1198871198861198871198861198872
1011986147520 119886119887119886
47520 119886211988731198862
1211986023760 119886
21198871198862
23760 1198861198873119886
12119861
47520 119887
47520 1198861198871198861198871198862
47520 1198871198861198872119886119887
1411986095040 119887119886119887119886
95040 1198861198872119886119887119886
1511986095040 119886119887
2
95040 119886119887119886119887
15119861 190080 1198861198871198861198872
1811986095040 119886119887
2119886119887
95040 119887119886119887119886119887
21119860 190080 11988711988621198871198862
2411986095040 119886119887
2119886
95040 119886211988721198862
2411986195040 119886
2119887119886
95040 119887119886
3011986095040 119886119887119886
2
95040 11988611988731198862
3152 119883 = 2119861 119905 = ((1198862119887)3119886119887)5 See Tables 38 and 39
316 119870 cong 1198621199002
3161 119883 = 2119860 119905 = 119886 See Table 40
3162 119883 = 2119861 119905 = 119886(119886119887119886119887119886)2119886(119886119887119886119887119886)7 See Table 41
3163 119883 = 2119862 119905 = (119887minus1(1198861198872)2119886119887minus1(1198861198872)4)5 See Table 42
Table 40
119883119862
|O| 119892
2119861 1260 1198871198861198872119886119887
2119862 1008 11988731198861198871198861198873
3119861 14336 119887119886119887
4119862 40320 119887
Table 41
119883119862
|O| 119892
2119860 15 (1198874119886)2119887311988611988721198861198874(119887119886119887)
2119886(119887119886119887)
711988611988711988611988731198861198872(119886119887)2
21198611680 119887
31198861198871198861198873
210 11988741198861198874119886119887119886119887
2119862 2520 (119886119887)5119886(119886119887)
2119886(119886119887)
3
3119861 35840 1198872119886119887
4119860 1920 11988711988611988731198861198872119886119887119886119887
4119862 20160 1198871198861198872119886119887
4119864 161280 119887119886119887
411986513440 119887119886119887119886119887119886119887119886119887
13440 119887311988611988711988611988721198861198872
5119861 344064 1198872
6119864 430080 119887
Table 42
119883119862
|O| 119892
2119860240 119886119887
4119886119887211988611988711988611988741198861198873119886119887311988611988741198861198873119886119887119886
45 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887411988611988741198861198872119886
2119861
720 1198872119886
240 11988741198861198874119886119887311988611988741198861198874119886119887211988611988731198861198872119886
180 119886119887211988611988731198861198872119886119887119886119887311988611988731198861198872
30 1198861198871198861198874119886
2119862
1440 1198861198871198861198873119886119887311988611988721198861198871198861198872119886
1440 119886119887311988611988741198861198873119886119887211988611988721198861198874119886
1440 1198872119886119887119886119887119886119887
311988611988741198861198873
1152 11988611988711988611988731198861198873119886119887311988611988741198861198872119886
360 11988611988711988611988711988611988731198861198872119886119887119886119887311988611988731198861198872
3119860 10240 1198861198874119886119887211988611988711988611988721198861198871198861198872119886
3119861
15360 11988611988731198861198872119886
7680 11988731198861198872
1024 1198861198872119886119887211988611988731198861198873119886
41198603840 119887
411988611988741198861198873119886119887411988611988741198861198874
1440 1198861198871198861198872119886119887211988611988741198861198872
4119861
11520 1198871198861198874119886119887411988611988731198861198873119886
5760 119886119887211988611988711988611988721198861198871198861198873119886
5760 11988611988721198861198874119886119887311988611988741198861198873119886
1440 11988611988741198861198871198861198872119886119887411988611988721198861198871198861198873119886
1440 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887119886
720 119886119887119886119887411988611988721198861198872119886119887211988611988731198861198873
720 1198872119886119887411988611988721198861198872119886119887411988611988731198861198874119886
Algebra 13
Table 42 Continued
119883119862
|O| 119892
4119862
11520 11988611988721198861198873119886
11520 1198861198873119886119887119886119887311988611988721198861198872
5760 119887311988611988741198861198872119886
5760 11988611988731198861198871198861198872
2880 11988611988711988611988741198861198872119886
1440 119887411988611988741198861198873119886119887211988611988721198861198873119886
1440 1198861198872119886119887311988611988731198861198872119886119887119886119887
720 11988611988711988611988711988611988711988611988721198861198871198861198873119886
720 119886119887311988611988741198861198872119886
4119863
23040 1198871198861198873119886119887119886119887211988611988731198861198872
11520 119886119887119886119887311988611988731198861198871198861198873
5760 119886119887411988611988711988611988741198861198872119886
2880 11988611988711988611988711988611988741198861198872119886119887411988611988741198861198873119886
2880 1198861198872119886119887119886119887119886119887
311988611988711988611988741198861198874119886
4119864
23040 1198871198861198871198861198871198861198872
23040 119886119887311988611988721198861198872
11520 1198861198873119886119887211988611988741198861198872
11520 119887311988611988711988611988731198861198872119886
5760 11988731198861198874119886
5760 1198861198871198861198872
5760 1198874119886119887311988611988721198861198871198861198873119886
4119865
23040 119886119887119886119887211988611988721198861198872
23040 1198873119886119887311988611988731198861198874119886
11520 119887411988611988741198861198872119886119887411988611988731198861198872119886
11520 119886119887311988611988741198861198873119886119887211988611988721198861198872
5760 1198872119886119887411988611988721198861198872119886119887211988611988731198861198872119886
5760 1198872119886119887211988611988731198861198873119886119887411988611988721198861198873119886
5760 1198861198873119886119887211988611988741198861198872119886119887311988611988731198861198872119886
5760 1198861198874119886119887119886119887119886119887
3119886119887211988611988721198861198873
5760 119886119887211988611988731198861198871198861198872119886119887211988611988731198861198873
5760 1198874119886119887411988611988721198861198872119886119887211988611988731198861198872119886
2880 119887311988611988721198861198871198861198872
2880 1198873119886119887411988611988731198861198872
1920 11988611988711988611988741198861198872119886119887211988611988741198861198873
1920 1198871198861198873119886119887311988611988711988611988721198861198874
4119866
46080 1198873119886119887311988611988721198861198873119886
46080 1198861198872119886119887311988611988721198861198872
23040 1198861198871198861198872119886119887211988611988731198861198872119886
23040 11988611988731198861198872119886119887311988611988731198861198874119886
5119860 73728 119886119887211988611988731198861198871198861198872119886
5119861184320 119887
21198861198872119886
122880 119887211988611988731198861198872
Table 42 Continued
119883119862
|O| 119892
6119860 92160 1198871198861198873119886119887411988611988731198861198872119886
6119861 122880 119887211988611988741198861198872119886
6119862
46080 1198872119886119887311988611988731198861198872119886
46080 119887311988611988711988611988721198861198872
15360 1198874119886119887211988611988731198861198873119886
15360 119886119887211988611988721198861198873119886119887
6119863
92160 119887411988611988711988611988721198861198874119886
23040 11988741198861198873119886119887411988611988721198861198872
15360 11988611988721198861198872119886119887119886
15360 119886119887411988611988731198861198873119886
6119864
92160 1198871198861198871198861198872
92160 1198871198861198874119886
46080 1198873119886119887211988611988731198861198872119886
46080 11988711988611988721198861198873119886
46080 11988721198861198871198861198874119886
46080 11988611988711988611988741198861198873
15360 119886119887211988611988711988611988741198861198871198861198872119886
6119865
92160 1198861198873119886119887119886
92160 11988611988741198861198872119886
46080 119887311988611988721198861198871198861198872119886
46080 1198861198874119886119887411988611988731198861198873
46080 1198861198873119886119887411988611988731198861198872
46080 11988721198861198872119886119887119886119887119886
7119860 737280 11988741198861198872
811986092160 119887
2119886119887119886
92160 11988611988741198861198873
8119861
92160 119886119887411988611988721198861198872119886119887119886
92160 119886119887411988611988731198861198873119886119887119886
46080 11988611988711988611988731198861198872119886
46080 119886119887311988611988721198861198874119886
23040 119887119886119887311988611988731198861198871198861198873
23040 11988721198861198874119886119887211988611988721198861198874
8119862
92160 1198873119886119887311988611988731198861198874
92160 1198872
92160 1198873
92160 119887119886119887211988611988721198861198872
8119863
92160 119887311988611988731198861198872
92160 11988611988721198861198873119886119887311988611988741198861198872
92160 119887411988611988731198861198873119886
92160 11988711988611988721198861198871198861198873119886
92160 119887311988611988721198861198872
14 Algebra
Table 42 Continued
119883119862
|O| 119892
8119864
92160 1198861198873119886
92160 1198861198872119886
46080 119886119887311988611988711988611988741198861198872
46080 119887311988611988711988611988741198861198872119886
46080 119886119887211988611988721198861198871198861198873
46080 1198872119886119887119886119887119886
46080 1198872119886119887411988611988731198861198873119886
46080 1198872119886119887211988611988731198861198873119886
46080 1198861198872119886119887211988611988731198861198873
46080 119886119887411988611988741198861198873
23040 1198861198874119886119887211988611988721198861198871198861198873
23040 11988741198861198874119886119887211988611988741198861198873
8119865
184320 11988611988741198861198871198861198873119886
184320 11988611988721198861198874119886119887119886
92160 119886119887119886119887311988611988711988611988721198861198872
92160 11988731198861198874119886119887411988611988731198861198872119886
9119860737280 119886119887
41198861198871198861198872
245760 119887211988611988741198861198872
10119860 368640 119887211988611988731198861198872119886
10119861368640 119886119887
21198861198872
368640 11988731198861198873119886
10119862368640 119886119887
4
368640 11988611988711988611988731198861198871198861198873119886
11119860 1474560 11988721198861198873119886
12119860122880 119887119886119887
21198861198871198861198874119886
122880 119887411988611988711988611988741198861198873
12119861368640 119887
211988611988711988611988731198861198872
184320 1198861198874119886119887211988611988741198861198872
12119862368640 119886119887
2119886119887211988611988731198861198872
184320 119886119887411988611988711988611988721198861198872
12119863
184320 119887411988611988711988611988721198861198872
184320 11988611988711988611988711988611988731198861198873119886
92160 119886119887411988611988721198861198873119886
92160 11988611988721198861198873119886119887119886
12119864368640 119887
311988611988721198861198874
368640 11988711988611988731198861198872
12119865
184320 11988721198861198871198861198872119886
184320 11988611988731198861198874119886
184320 11988611988711988611988731198861198873
184320 1198861198871198861198872119886
92160 1198861198872119886119887119886
92160 11988611988741198861198873119886
12119866 368640 11988711988611988741198861198872119886
Table 42 Continued
119883119862
|O| 119892
12119867
184320 1198874119886119887311988611988731198861198872
184320 119886119887311988611988721198861198873119886
184320 11988611988711988611988711988611988741198861198872119886
184320 1198861198874119886119887311988611988741198861198872119886
184320 119887311988611988721198861198872119886119887
184320 119886119887211988611988731198861198872119886
14119860 737280 1198861198874119886
15119860737280 119887
4119886
737280 1198871198861198872119886
16119860
368640 11988611988731198861198872
368640 11988611988721198861198871198861198874
368640 11988731198861198872119886
368640 11988711988611988741198861198873119886
18119860 737280 11988731198861198873
20119860737280 119886119887
41198861198872
737280 1198873119886119887119886
20119861737280 119886119887
41198861198874119886
737280 119886119887119886119887119886
24119860737280 119886119887
2
737280 1198873119886
24119861737280 119887
4
737280 119887
28119860737280 119886119887119886119887
31198861198872
737280 119886119887411988611988721198861198872
30119860737280 119887
21198861198874
737280 1198871198861198873
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R H Bing ldquoA homeomorphism between the 3-sphere andthe sum of two solid horned spheresrdquo Annals of MathematicsSecond Series vol 56 pp 354ndash362 1952
[2] D Montgomery and L Zippin ldquoExamples of transformationgroupsrdquo Proceedings of the American Mathematical Society vol5 pp 460ndash465 1954
[3] J W Morgan and H Bass Eds The Smith Conjecture vol 112of Pure and Applied Mathematics Academic Press OrlandoFla USA 1984 Papers presented at the symposium held atColumbia University New York NY USA 1979
[4] C M Edwards and G T Ruttimann ldquoInvolutive and Peircegradings in JBWlowast-triplesrdquo Communications in Algebra vol 31no 6 pp 2819ndash2848 2003
Algebra 15
[5] J Gornicki and B E Rhoades ldquoA general fixed point theoremfor involutionsrdquo Indian Journal of Pure and Applied Mathemat-ics vol 27 no 1 pp 13ndash23 1996
[6] S Montgomery Fixed Rings of Finite Automorphism Groupsof Associative Rings vol 818 of Lecture Notes in MathematicsSpringer Berlin Germany 1980
[7] T A Springer ldquoThe classification of involutions of simplealgebraic groupsrdquo Journal of the Faculty of Science University ofTokyo Section IA Mathematics vol 34 no 3 pp 655ndash670 1987
[8] T A Springer ldquoSome results on algebraic group s with invo-lutionsrdquo in Algebraic Groups and Related Topics (KyotoNagoya1983) vol 6 of Advanced Studies in Pure Mathematics pp 525ndash543 North-Holland Amsterdam The Netherlands 1985
[9] W Feit and J GThompson ldquoSolvability of groups of odd orderrdquoPacific Journal of Mathematics vol 13 pp 775ndash1029 1963
[10] R Brauer and K A Fowler ldquoOn groups of even orderrdquo Annalsof Mathematics vol 62 no 3 pp 565ndash583 1955
[11] D Gorenstein The Classification of Finite Ssimple Groups Vol1 Groups of Noncharacteristic 2 Type The University Series inMathematics Plenum Press New York NY USA 1983
[12] D Gorenstein R Lyons and R Solomon The Classification ofthe Finite Simple Groups vol 40 of Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 1994
[13] C Bates and P Rowley ldquoInvolutions in Conwayrsquos largest simplegrouprdquo LMS Journal of Computation andMathematics vol 7 pp337ndash351 2004
[14] P Rowley and P Taylor ldquoPoint-line collinearity graphs of twosporadic minimal parabolic geometriesrdquo Journal of Algebra vol331 pp 304ndash310 2011
[15] P Rowley and P Taylor ldquoInvolutions in Jankorsquos simple group 1198694rdquoLMS Journal of Computation and Mathematics vol 14 pp 238ndash253 2011
[16] P Taylor ldquoInvolutions in Fischerrsquos sporadic groupsrdquo preprinthttpeprintsmamanacuk1622
[17] The GAP Group GAP-Groups Algorithms Programming Ver-sion 43 2002 httpwwwgap-systemorg
[18] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I The user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997
[19] R Wilson P Walsh J Tripp et al ldquoAtlas o f finite grouprepresentationsrdquo httpbrauermathsqmulacukAtlasv3
[20] J H Conway R T Curtis S P Norton R A Parker and R AWilson Atlas of Finite Groups Clarendon Oxford UK 1985
[21] R AWilson ldquoStandard generators for sporadic simple groupsrdquoJournal of Algebra vol 184 no 2 pp 505ndash515 1996
[22] D Gorenstein Finite Groups Chelsea Publishing New YorkNY USA 2nd edition 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Algebra
Table 17
119883119862
|O| 119892
2119860 105 119886
2119861 28 119887119886
3119860 336 1198872
4119861 630 119887
Table 18
119883119862
|O| 119892
211986060 119886119887
31198861198871198861198872119886
5 11988611988731198861198872119886
2119861 60 1198872119886
3119860240 119886
160 11988611988731198861198871198861198872119886119887119886
4119860 120 1198871198861198873119886119887119886
4119861
240 119886119887
120 1198871198861198872119886119887119886119887
30 1198861198872119886
4119862120 119886119887
21198861198872119886
120 1198861198873119886119887119886
5119860192 119886119887119886119887
3
192 11988711988611988721198861198873
5119861 320 1198874119886119887
5119862 1920 11988721198861198872
6119860480 119886119887
2119886119887
480 1198871198861198872119886
Table 19
119883119862
|O| 119892
6119861480 119887
4119886119887119886119887
480 11988721198861198872119886
7119860 1920 119887
8119860480 119887119886119887119886119887
2
480 11988711988611988721198861198872119886119887
10119860960 119887
211988611988721198861198872119886
960 119887119886119887
10119861 960 1198861198872
11119860 1920 1198872119886119887119886
11119861 1920 119887119886119887119886119887
12119860960 119887119886
960 1198873
15119860 1920 1198871198861198873
20119860960 119887119886119887119886
960 1198872119886119887
20119861960 119887
21198861198873119886
960 1198874119886119887119886
Table 20
119883119862
|O| 119892
2119860120 119887(119887119886)
3119887(119887119886)119887(119887119886)
5
10 (119887119886)119887(119887119886)4119887(119887119886)
2
3119860 192 119887(119887119886)6
3119861 640 (119887119886)119887(119887119886)2
4119860480 119887(119887119886)
5119887(119887119886)
80 (119887119886)3119887(119887119886)
3119887(119887119886)
4
5119860960 (119887119886)
3119887(119887119886)
3119887(119887119886)
2
320 (119887119886)2119887(119887119886)
2
5119861960 (119887119886)119887(119887119886)
3119887(119887119886)
4
320 (119887119886)119887(119887119886)3
6119860 960 (119887119886)119887(119887119886)6
8119860960 (119887119886)
4119887(119887119886)
2
960 (119887119886)119887(119887119886)5
9119860 1920 119887(119887119886)2119887(119887119886)
2
9119861 1920 (119887119886)
9119862 1920 119887(119887119886)2
10119860960 (119887119886)
4119887(119887119886)
960 119887(119887119886)4119887(119887119886)119887(119887119886)
10119861960 119887(119887119886)119887(119887119886)119887(119887119886)119887(119887119886)
2
960 119887(119887119886)5
12119860960
960
1003816100381610038161003816100381610038161003816100381610038161003816
(119887119886)2119887(119887119886)
4
(119887119886)6
15119860 1920 (119887119886)2119887(119887119886)
15119861 1920 119887(119887119886)3
17119860 1920 (119887119886)2
17119861 1920 (119887119886)2119887(119887119886)
3
Table 21
119883119862
|O| 119892
2119860 153 1198861198872119886119887119886119887119886119887
3119860102
102
1003816100381610038161003816100381610038161003816100381610038161003816
1198872
119887
3119861 272 1198861198871198861198872119886
4119860612 119887119886119887119886119887119886119887
2119886
306 1198871198861198872119886119887119886119887119886
6119860
612
612
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198872119886119887119886
1198871198861198871198861198872119886
306
306
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198872119886119887119886119887119886
119887119886119887119886
81198601224 119886119887119886119887119886119887
1224 1198872119886119887119886119887119886
9119860 816 11988721198861198871198861198872119886119887
9119861 816 1198871198861198871198861198872119886119887
9119862 816 1198871198861198872
121198601224
1224
1003816100381610038161003816100381610038161003816100381610038161003816
119886119887
1198861198872119886119887119886
17119860 2448 119887119886
17119861 2448 119886
19119860 2448 1198861198872119886119887
19119861 2448 1198872119886119887119886
Algebra 7
Table 22
119883119862
|O| 119892
2119860
168 1198872119886119887119886119887119886119887
2119886119887
14 119886119887119886
14 1198861198871198861198872119886119887
2119861 84 1198872119886119887
3119860 896 119887
3119861 128 119887211988611988721198861198871198861198872119886119887119886119887
Table 23
119883119862
|O| 119892
4119860112 119887119886119887119886119887119886119887
112 1198872119886119887211988611988721198861198872
4119861
1344 1198861198872119886119887119886
672 11988611988721198861198871198861198872119886119887
672 11988721198861198871198861198872119886119887119886
5119860 1792 119887119886119887119886
6119860 5376 119886
Table 24
119883119862
|O| 119892
211986060 119887
2119886119887119886119887211988611988721198861198871198861198872119886119887119886119887119886119887
2119886119887
15 11988711988611988721198861198872119886119887119886119887119886119887
2
2119861
120 119887119886119887211988611988711988611988721198861198871198861198872
80 119887211988611988711988611988721198861198871198861198872119886119887
2 (119887119886)8(1198872119886)21198871198861198872(119886119887)2(1198861198872)2(119886119887)3(1198861198872119886119887)2119887
3119860 320 119887211988611988721198861198871198861198872119886119887119886119887119886119887
3119861 960 11988721198861198871198861198872119886119887
4119860120 119887119886119887
21198861198872
120 1198871198861198871198861198872
4119861
480 1198872119886119887119886119887119886119887
2119886119887
240 1198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
240 1198872119886119887119886119887119886119887119886119887119886119887119886119887
4119862
960 1198871198861198872119886119887
960 11988721198861198871198861198872
160 1198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
21198861198872
160 1198872119886119887119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
5119860 1920 119887119886119887
6119860 1920 1198871198861198871198861198871198861198872
61198611920 119887119886119887
2
1920 1198872119886119887
101198601920 119887
21198861198871198861198872119886119887119886119887
1920 1198872119886119887119886119887119886119887119886119887
2
11119860 7680 119887119886119887119886119887
121198613840 119887
3840 1198872
3102 119883 = 2119861 119905 = 119886 See Table 24
311 119870 cong 119872119888119871
3111119883 = 2119860 119905 = 119886 See Table 25
3112119883 = 2119861 119905 = 119888 See Table 26
Table 25
119883119862
|O| 119892
2119860 210 119887211988611988731198861198872
3119861 2240 1198871198861198871198861198872
4119860 5040 1198872119886119887
5119861 8064 119887
6119861 6720 1198872
Table 26
119883119862
|O| 119892
2119860 165 11988611988721198861198871198861198872119886119887
3119860 220 1198872119886119887311988611988721198861198871198861198872119886
3119861660 119886119887
21198861198872
220 1198872119886119887119886119887119886119887
3
41198601980 119887
21198861198873119886119887119886119887119886
990 119886
5119860792 119887
211988611988721198861198873
792 1198872119886119887119886119887119886119887
2
5119861 7920 1198872
6119860 1980 119886119887
6119861
1980 1198871198861198872119886119887119886119887
1980 1198861198872119886119887
660
660
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198873
11988711988611988731198861198874119886119887
71198603960 119887119886119887119886119887
2119886
3960 1198871198861198872
71198613960 119887119886119887
3119886
3960 1198872119886
81198603960 119887
21198861198872119886
3960 11988611988731198861198873
9119860 2640 1198872119886119887119886
9119861 2640 1198861198871198861198872119886119887
101198603960 119887119886119887119886119887
3960 119886119887119886119887119886
121198603960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198873
1198871198861198873
141198603960 119886119887
21198861198873
3960 1198871198861198872119886119887
141198613960 119887119886119887119886119887119886
3960 1198872119886119887119886119887119886
151198603960 119887119886119887
3960 1198874
151198613960 119886119887119886119887
3960 119887
301198603960 119887119886
3960 1198871198861198872119886
301198613960 119887
41198861198872119886
3960 1198861198872
8 Algebra
Table 27
119883119862
|O| 119892
2119860240 119887119886119887119886119887119886119887
2 1198875(1198861198876)2(119886119887)31198875119886(11988721198861198873)2
2119861 315 119887211988611988711988611988721198861198872119886119887
3119860 1344 1198872119886119887
3119861 960 1198871198861198874119886119887
4119860480 119887
31198861198871198861198872119886119887
480 11988721198861198871198861198873119886119887
41198612520
2520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887
1198874119886119887
5119860 2688 1198873
6119860 13440 119887
Table 28
119883119862
|O| 119892
2119860 84 (119886119887)21198875(119886119887)2
2119861
168
168
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)2
(119886119887)1198874(119886119887)119887
2(119886119887)119887
2(119886119887)119887
4
14
14
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198874(119886119887)119887
4(119886119887)119887
2(119886119887)119887
2
1198872(119886119887)119887
2(119886119887)21198874(119886119887)119887
2(119886119887)119887
2
3119860 896 1198874(119886119887)119887
2
3119861 2688 1198872
4119861
1344
1344
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)119887
2(119886119887)119887
2
1198872(119886119887)119887
4(119886119887)21198874
112
112
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)1198872(119886119887)41198874(119886119887)2
(119886119887)31198874(119886119887)41198872
4119862
1344
1344
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)1198874(119886119887)119887
4
1198874(119886119887)119887
2(119886119887)119887
2(119886119887)31198872
672
672
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198872(119886119887)3
1198872(119886119887)41198872(119886119887)2
672
672
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)41198872(119886119887)2
(119886119887)4119887(119886119887)
2
5119860 5376 (119886119887)3
6119860 5376 (119886119887)1198872(119886119887)119887
2(119886119887)4
61198615376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198872(119886119887)
(119886119887)1198872(119886119887)
7119862 3072 1198872(119886119887)119887
2(119886119887)119887
2(119886119887)2
81198605376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)21198872(119886119887)21198872
(119886119887)1198872(119886119887)21198872(119886119887)
101198605376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)2
1198872(119886119887)119887
2(119886119887)119887
2
1211986110752
10752
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)2
1198874(119886119887)2
15119860 21504 1198872(119886119887)3
17119860 21504 (119886119887)1198872(119886119887)2
17119861 21504 (119886119887)
21119860 21504 (119886119887)1198872(119886119887)119887
2
21119861 21504 1198872(119886119887)
Table 29
119883119862
|O| 119892
2119860105 (119886119887)119887(119886119887)
3119887(119886119887)
3
63 119887611988611988731198861198873119886119887211988611988761198861198872119886119887611988611988761198861198874119886
2119861 315 (1198861198872)2(119887119886)21198872119886
3119860630 (119886119887)119887(119886119887)119887(119886119887)119887
126 119887(119886119887)21198876
3119861 2520 (119886119887)1198876
41198601260 (119886119887)
21198875(119886119887)
1260 (119886119887)119887(119886119887)1198873
4119861
945
945
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)21198872(119886119887)119887
119887(119886119887)119887(119886119887)1198872(119886119887)119887
315
315
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119886119887)5119887
119887(119886119887)31198872(119886119887)119887
41198621890
1890
1003816100381610038161003816100381610038161003816100381610038161003816
1198873(119886119887)2
(119886119887)1198872(119886119887)2119887
5119860 2520 (119886119887)3119887(119886119887)
6119860
3780 (119886119887)31198872(119886119887)2
1890 119887(119886119887)119887(119886119887)
1890 119887(119886119887)119887(119886119887)119887(119886119887)3
1260 119887(119886119887)1198872(119886119887)119887(119886119887)
2
6119861 7560 (119886119887)
71198601890 (119886119887)119887(119886119887)
41198872
90 119887(119886119887)1198873(119886119887)2119887
71198611890 119887
5(119886119887)119887(119886119887)
90 1198875(119886119887)119887
5(119886119887)119887
2(119886119887)119887
5
7119862 2160 119887(119886119887)119887(119886119887)4
71198637560 (119886119887)119887(119886119887)
4
1080 (119886119887)1198873(119886119887)
71198647560 (119886119887)
3119887(119886119887)
2
1080 (119886119887)1198874(119886119887)2119887(119886119887)
8119860
3780
3780
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)2119887(119886119887)
2
(119886119887)4
3780
3780
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)3119887(119886119887)119887(119886119887)119887
(119886119887)119887(119886119887)119887(119886119887)2
10119860 7560 (119886119887)3
121198607560 119887(119886119887)119887
7560 119887(119886119887)1198872
121198617560
7560
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119886119887)2119887
119887
14119860
3780 (119886119887)1198873(119886119887)2
3780 (119886119887)21198872(119886119887)
1890 (119886119887)1198872(119886119887)
1890 119887(119886119887)3119887
14119861
3780 (119886119887)119887(119886119887)2119887(119886119887)
3780 (119886119887)1198874
1890 (119886119887)21198872(119886119887)119887(119886119887)
2
1890 119887(119886119887)119887(119886119887)2
Algebra 9
Table 29 Continued
119883119862
|O| 119892
141198627560 119887(119886119887)
7560 119887(119886119887)3
141198637560 119887
3(119886119887)
7560 119887(119886119887)2
15119860 15120 (119886119887)1198872
21119860 15120 (119886119887)4119887(119886119887)
21119861 15120 (119886119887)119887
21119862 15120 (119886119887)3119887
21119863 15120 1198872
281198607560 119887
2(119886119887)119887
2
7560 (119886119887)2
281198617560 119887
4(119886119887)3
7560 (119886119887)119887(119886119887)2
Table 30
119883119862|O| 119892
2119860
640 11988611988731198861198872119886119887119886
480 11988611988721198861198872119886119887119886119887119886119887119886119887119886119887
21198861198872119886
240 11988611988721198861198873119886119887311988611988731198861198873119886119887211988611988711988611988721198861198873119886119887119886119887119886119887119886119887
31198861198873
30 1198861198872119886119887119886119887119886119887211988611988721198861198873119886119887311988611988711988611988711988611988721198861198873119886119887211988611988721198861198873119886119887211988611988711988611988711988611988731198863119860 10240 119886119887
3119886119887119886119887119886119887119886119887119886119887
2119886
4119860 5120 119886119887119886119887119886119887119886
4119862
7680 119886119887119886119887211988611988721198861198872119886119887119886
3840 119886119887211988611988731198861198871198861198872119886119887119886
3840 119886119887119886119887211988611988711988611988731198861198872119886
411986315360 119886119887119886
7680 1198861198872119886119887211988611988711988611988721198861198871198861198872119886
5119860 12288 11988611988731198861198871198861198872119886119887119886119887119886119887119886
5119861 4096 119886119887119886119887119886
611986061440 119886119887
21198861198871198861198872119886119887119886
30720 119886119887119886119887119886119887119886119887119886119887119886
7119860 61440 119886
811986261440 119886119887119886119887
21198861198872119886
61440 11988611988721198861198872119886119887119886
10119860 61440 11988611988721198861198872119886119887119886119887119886119887119886
12119860122880 1198861198872119886119887119886
13119860 61440 1198861198873119886119887119886
312 119870 cong 119867119890
3121119883 = 2119860 119905 = 119886 See Table 27
3122119883 = 2119861 119905 = (1198861198872)6 See Table 28
3123 119883 = 2119862 119905 = 1198893 See Table 29
313 119870 cong 119877119906
3131 119883 = 2119860 119905 = 1198872 See Table 30
3132 119883 = 2119861 119905 = 119886 See Table 31
314 119870 cong 119878119906119911
3141119883 = 2119860 119905 = ((119886119887)21198861198872)6 See Table 32
Table 31
119883119862 |O| 119892
2119860 455 119887119886119887311988611988721198861198873119886119887
2119861
455 11988711988611988731198861198872119886119887119886119887119886119887119886119887119886119887
455 11988711988611988711988611988711988611988711988611988711988611988721198861198873119886119887
1 1198873119886119887311988611988731198861198872119886119887119886119887119886119887
211988611988721198861198873119886119887119886119887211988611988721198861198872
1 1198871198861198872119886119887119886119887311988611988711988611988721198861198872119886119887119886119887211988611988711988611988731198861198873119886119887119886119887119886119887
31198861198871198861198872119886119887119886119887211988611988731198861198871198861198873
3119860 5824 1198872119886119887119886119887119886119887119886119887119886119887
4119860 3640 (119886119887)10119886(119886119887)15119886((119886119887)
5119886)2
4119862 3640 1198873119886119887119886119887119886119887
21198861198872
4119863 3640 119887311988611988711988611988721198861198873119886119887
51198605824 119887
311988611988711988611988721198861198871198861198872
5824 119887211988611988731198861198872119886119887
5119861 29120 1198872119886119887119886119887
6119860 29120 1198872119886119887119886119887119886119887
2119886119887
7119860
29120 11988721198861198872
29120 119887
29120 1198872
1011986029120 119887119886119887119886119887
2119886119887119886119887
29120 119887119886119887119886119887
1011986129120 119887119886119887119886119887
31198861198873119886119887
29120 11988711988611988731198861198873119886119887119886119887
12119860 58240 1198873119886119887119886119887119886119887
2119886119887
13119860
29120 119887119886119887
29120 119887119886119887119886119887119886119887
29120 11988731198861198872119886119887
1411986029120 119887
311988611988721198861198872119886119887119886119887
29120 11988711988611988721198861198872119886119887119886119887119886119887
1411986129120 119887
21198861198872119886119887119886119887119886119887
29120 11988711988611988711988611988711988611988721198861198872
1411986229120 119887119886119887
2119886119887119886119887119886119887119886119887
2
29120 1198872119886119887119886119887119886119887119886119887
2119886119887
15119860 116480 11988731198861198872119886119887119886119887119886119887
2011986058240 119887
31198861198871198861198872
58240 11988721198861198873119886119887
2611986029120 119887119886119887
2
29120 1198872119886119887
2611986129120 119887119886119887
21198861198872
29120 11988721198861198872119886119887
2611986229120 119887
3119886119887119886119887119886119887
29120 11988711988611988711988611988721198861198872
29119860 116480 1198871198861198872119886119887
29119861 116480 1198873119886119887119886119887
Table 32
119883119862
|O| 119892
2119860360 119886
54 ((119886119887)2119886)4119886((119886119887)
2119886)3
3119861 5120 11988721198861198872119886119887119886119887119886119887119886119887119886119887119886
3119862 9216 119887119886119887119886
4119860 1728 119887211988611988721198861198872119886119887119886119887119886119887
4119862 17280 1198872119886119887
5119861 55296 1198872119886119887119886119887119886
6119863 46080 119887
10 Algebra
Table 33119883119862|O| 119892
2119860 315 1198871198861198872119886119887119886119887119886119887
21198861198872119886119887211988611988721198861198872119886119887119886119887119886119887
2119886119887
2119861
630 1198872119886119887119886119887119886119887119886119887
2119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
21198861198872119886119887119886119887211988611988721198861198872119886119887
560 119887119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
2 (1198871198861198872119886)2(1198872119886)2(119887119886(1198872119886)2)2(1198872119886119887119886)2((1198872119886)21198872119886119887119886)21198871198861198872(119886119887)2
3119860 224 (1198871198861198872119886)3119887(119887119886)3(1198872119886)2(119887119886)5119887(119887119886)4(1198872119886)2119887
3119861 2240 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887
3119862 20160 11988711988611988721198861198871198861198872119886119887
41198602520 119887119886119887119886119887119886119887
2119886119887211988611988721198861198872119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886119887119886119887
4119861
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
2520 11988711988611988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
21198861198872119886119887
1260 119887211988611988721198861198871198861198872119886119887119886119887
4119862
5040 11988711988611988721198861198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
2520 11988711988611988711988611988711988611988721198861198872119886119887119886119887119886119887
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
4119863
10080 11988711988611988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
10080 11988711988611988711988611988721198861198872119886119887119886119887211988611988721198861198872119886119887
1120 (119886119887)4((119886119887)4119886)4119886(119886119887)8((119886119887)4119886)2
1120 (119886(1198872119886)4)2(1198872119886)8119886(119886(1198872119886)4)4(1198872119886)4
5119860 20160 119887119886119887119886119887
5119861 40320 1198872119886119887119886119887119886119887
2119886119887
6119860 10080 1198871198861198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
6119863 20160 11988711988611988721198861198872119886119887
611986440320 119887
2119886119887119886119887119886119887
2119886119887119886119887119886119887
40320 1198872119886119887119886119887211988611988721198861198872119886119887119886119887119886119887
7119860 80640 1198872119886119887119886119887119886119887119886119887
811986020160 119887119886119887
2119886119887119886119887211988611988721198861198872119886119887
20160 1198871198861198872119886119887211988611988721198861198871198861198872119886119887
8119861
20160 1198871198861198872119886119887119886119887119886119887119886119887
211988611988721198861198871198861198872119886119887
20160 11988711988611988711988611988721198861198872119886119887
20160 11988711988611988721198861198872119886119887119886119887
20160 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887119886119887
2119886119887
10080 11988721198861198872119886119887
10080 1198872119886119887119886119887
8119862
20160 11988721198861198871198861198872119886119887119886119887119886119887
20160 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
2119886119887
20160 1198872119886119887211988611988721198861198871198861198872119886119887
20160 1198871198861198872119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
10119860
40320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
119887211988611988711988611988721198861198871198861198872119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886119887
20160 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887
20160 11988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
1011986140320 119887119886119887119886119887119886119887119886119887
2119886119887
40320 1198871198861198872119886119887119886119887119886119887119886119887
11119860 161280 11988721198861198871198861198872119886119887119886119887
1211986040320 119887
2119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
20160 119887211988611988711988611988721198861198872119886119887119886119887119886119887
2119886119887
1211986140320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886119887
11988721198861198872119886119887211988611988721198861198872119886119887
12119862 40320 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
1211986380640 119887119886119887
2119886119887119886119887119886119887
80640 1198871198861198871198861198871198861198872119886119887
13119860 161280 119887119886119887
13119861 161280 11988721198861198872119886119887119886119887
14119860 80640 119887211988611988711988611988721198861198872119886119887119886119887
15119860 161280 119887
15119861 161280 1198872119886119887
15119862 161280 119887211988611988721198861198871198861198872119886119887119886119887119886119887
Table 33 Continued119883119862
|O| 119892
2011986080640
80640
1003816100381610038161003816100381610038161003816100381610038161003816
119887119886119887211988611988721198861198872119886119887119886119887
119887119886119887119886119887211988611988721198861198872119886119887
21119860 161280 119887119886119887211988611988721198861198872119886119887119886119887119886119887
21119861 161280 119887119886119887211988611988721198861198872119886119887
2411986080640 119887119886119887119886119887
2119886119887
80640 1198871198861198872119886119887119886119887
Table 34
119883119862
|O| 119892
2119860 315 ((119886119887)3119886)2119886((119886119887)
3119886)4119886(119886119887)
3((119886119887)3119886)2119886
2119861 1800 1198861198871198861198871198861198872
3119860 200 119887211988611988721198861198871198861198872119886119887119886119887
3119862 16800 11988611988721198861198872119886119887119886119887119886
41198603150 119887119886119887
21198861198871198861198872119886
630 1198871198861198872119886119887119886119887119886119887119886119887119886119887
2119886
4119862 12600 1198871198861198872119886119887119886
5119861 20160 119886119887119886
6119860 12600 1198861198871198861198871198861198872119886119887
6119864 100800 119887119886
7119860 100800 119887
811986025200 119886119887119886119887119886119887
25200 119887119886119887119886119887119886119887
12119860 50400 119886
Table 35
119883119862
|O| 119892
2119860 495 (119886119887)5119886 (119886119887)
2(119886 (119886119887)
5)4
(119886119887)5
2119861 792 11988721198861198871198861198872119886119887119886119887119886119887
2119886119887119886119887
3119860 264 119886(119886119887)9((119886119887)9119886)4119886(119886119887)
9((119886119887)9119886)2
31198615280 119886(119886119887)
6((119886119887)3119886)4
1760 (119886((119886119887)3119886)2)4119886
3119862 2640 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886
4119860
2970 11988711988611988711988611988721198861198871198861198872119886119887119886
495
495
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886
4119861 11880 119887119886119887211988611988721198861198871198861198872119886119887119886
4119862 11880 1198872119886119887
5119860 19008 119887119886119887119886
5119861 47520 1198861198872119886119887119886119887119886
6119860
11880 119886119887119886
3960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198871198861198872119886
11988611988711988611988721198861198872119886
611986115840 119886119887
2119886119887119886119887
15840 1198861198872119886119887119886119887119886119887119886119887119886119887
611986215840 119887
21198861198872119886119887119886
15840 1198872119886119887119886119887119886119887119886119887119886119887
21198861198872
611986347520 119886119887
2119886119887119886119887119886119887119886
15840 1198861198872119886119887119886119887119886119887
2119886119887119886
Algebra 11
Table 35 Continued
119883119862
|O| 119892
6119864 15840 119886119887119886119887119886119887119886119887211988611988711988611988721198861198872119886119887
7119860 47520 1198871198861198872119886119887119886
811986147520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886
119886119887119886119887119886119887
911986095040 119887119886119887119886119887119886119887
31680 119887119886119887119886119887211988611988721198861198871198861198872119886
911986195040 119887
2119886119887119886119887119886
31680 1198861198872119886119887119886119887119886119887119886119887119886119887119886
10119860 95040 119886119887
1011986147520 119887119886119887119886119887
21198861198872119886
47520 11988711988611988721198861198872119886119887119886
11119860 190080 119886
12119860 47520 1198872119886119887119886
12119861
47520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198872119886119887119886119887119886119887
2
119886119887119886119887119886119887119886119887119886
15840
15840
1003816100381610038161003816100381610038161003816100381610038161003816
11988721198861198872119886119887119886119887
119887119886119887119886119887119886
1211986247520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198872119886119887119886
1198861198871198861198871198861198871198861198871198861198871198861198872
12119864
47520 11988721198861198872119886
47520 119886119887119886119887
47520 119887119886119887119886119887119886119887119886119887
47520 1198872119886119887119886119887119886119887119886
1411986047520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198871198861198872119886119887119886119887
119887
15119862 190080 119886119887119886119887119886
1811986095040 119887119886119887
2119886119887119886119887119886
95040 11988611988721198861198872119886119887119886119887
1811986195040 119887
21198861198872119886119887119886119887119886
95040 1198871198861198871198861198871198861198872119886
2011986095040
95040
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198872119886119887119886
119887119886
Table 36
119883119862
|O| 119892
2119860 630 119886119887211988621198872119886
3119861 8960 11988611988721198862
3119862 1920 11988621198871198862
Table 37
119883119862
|O| 119892
4119861 30240 1198861198872119886
5119861 48384 1198861198871198862
6119862 80640 119886
Table 38
119883119862
|O| 119892
2119860 495 119887119886119887119886211988711988621198871198861198872119886119887
2119861396 119886119887
311988611988721198862119887119886
396 11988621198873119886119887211988621198871198862
31198601320 119887119886
211988711988621198872119886119887
440 11988611988721198861198871198861198872119886119887119886119887
31198612640 119886
211988721198861198871198861198873119886
1584 119886119887119886119887119886211988731198862
3119862 7920 11988711988611988721198861198872119886119887
4119860 2970 1198871198861198871198861198871198861198872119886119887119886
4119861
11880 1198862119887
5940 1198872
2970 1198871198861198871198861198872119886119887
51198609504 119887
21198861198871198862
9504 119887119886211988731198862119887119886
5119861 47520 11988621198872119886
611986011880 119887119886
211988721198861198872119886
3960 119886119887211988621198871198861198871198862119887119886
611986115840 119886
2
15840 119886
6119862
23760 1198871198861198871198861198873119886
23760 11988711988621198872119886119887119886
15840 1198871198861198873119886119887119886
611986423760 119887
2119886
23760 11988621198872
711986095040 119887119886
2
31680 1198871198861198872119886119887119886
811986023760 119887119886119887119886119887
2119886
23760 1198872119886119887119886119887119886119887
811986123760 119886119887119886119887
2119886
23760 11988721198862119887119886
3142 119883 = 2119861 119905 = 119886 See Table 33
3143 119883 = 2119862 119905 = 119888 See Table 34
3144 119883 = 2119863 119905 = (119888(119888119889)12)11 See Table 35
315 119870 cong 1198621199003
3151 119883 = 2119860 119905 = 1198872 See Tables 36 and 37
12 Algebra
Table 39
119883119862
|O| 119892
8119862
23760 11988611988721198861198872
23760 1198872119886211988721198862
23760 1198872119886119887119886
23760 11988711988621198871198861198873
911986031680 119887119886119887
21198861198873
31680 119886119887119886119887119886119887119886119887
9119861 63360 1198861198871198861198871198861198872
1011986047520 119886119887119886
21198872
47520 1198871198861198871198861198872
1011986147520 119886119887119886
47520 119886211988731198862
1211986023760 119886
21198871198862
23760 1198861198873119886
12119861
47520 119887
47520 1198861198871198861198871198862
47520 1198871198861198872119886119887
1411986095040 119887119886119887119886
95040 1198861198872119886119887119886
1511986095040 119886119887
2
95040 119886119887119886119887
15119861 190080 1198861198871198861198872
1811986095040 119886119887
2119886119887
95040 119887119886119887119886119887
21119860 190080 11988711988621198871198862
2411986095040 119886119887
2119886
95040 119886211988721198862
2411986195040 119886
2119887119886
95040 119887119886
3011986095040 119886119887119886
2
95040 11988611988731198862
3152 119883 = 2119861 119905 = ((1198862119887)3119886119887)5 See Tables 38 and 39
316 119870 cong 1198621199002
3161 119883 = 2119860 119905 = 119886 See Table 40
3162 119883 = 2119861 119905 = 119886(119886119887119886119887119886)2119886(119886119887119886119887119886)7 See Table 41
3163 119883 = 2119862 119905 = (119887minus1(1198861198872)2119886119887minus1(1198861198872)4)5 See Table 42
Table 40
119883119862
|O| 119892
2119861 1260 1198871198861198872119886119887
2119862 1008 11988731198861198871198861198873
3119861 14336 119887119886119887
4119862 40320 119887
Table 41
119883119862
|O| 119892
2119860 15 (1198874119886)2119887311988611988721198861198874(119887119886119887)
2119886(119887119886119887)
711988611988711988611988731198861198872(119886119887)2
21198611680 119887
31198861198871198861198873
210 11988741198861198874119886119887119886119887
2119862 2520 (119886119887)5119886(119886119887)
2119886(119886119887)
3
3119861 35840 1198872119886119887
4119860 1920 11988711988611988731198861198872119886119887119886119887
4119862 20160 1198871198861198872119886119887
4119864 161280 119887119886119887
411986513440 119887119886119887119886119887119886119887119886119887
13440 119887311988611988711988611988721198861198872
5119861 344064 1198872
6119864 430080 119887
Table 42
119883119862
|O| 119892
2119860240 119886119887
4119886119887211988611988711988611988741198861198873119886119887311988611988741198861198873119886119887119886
45 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887411988611988741198861198872119886
2119861
720 1198872119886
240 11988741198861198874119886119887311988611988741198861198874119886119887211988611988731198861198872119886
180 119886119887211988611988731198861198872119886119887119886119887311988611988731198861198872
30 1198861198871198861198874119886
2119862
1440 1198861198871198861198873119886119887311988611988721198861198871198861198872119886
1440 119886119887311988611988741198861198873119886119887211988611988721198861198874119886
1440 1198872119886119887119886119887119886119887
311988611988741198861198873
1152 11988611988711988611988731198861198873119886119887311988611988741198861198872119886
360 11988611988711988611988711988611988731198861198872119886119887119886119887311988611988731198861198872
3119860 10240 1198861198874119886119887211988611988711988611988721198861198871198861198872119886
3119861
15360 11988611988731198861198872119886
7680 11988731198861198872
1024 1198861198872119886119887211988611988731198861198873119886
41198603840 119887
411988611988741198861198873119886119887411988611988741198861198874
1440 1198861198871198861198872119886119887211988611988741198861198872
4119861
11520 1198871198861198874119886119887411988611988731198861198873119886
5760 119886119887211988611988711988611988721198861198871198861198873119886
5760 11988611988721198861198874119886119887311988611988741198861198873119886
1440 11988611988741198861198871198861198872119886119887411988611988721198861198871198861198873119886
1440 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887119886
720 119886119887119886119887411988611988721198861198872119886119887211988611988731198861198873
720 1198872119886119887411988611988721198861198872119886119887411988611988731198861198874119886
Algebra 13
Table 42 Continued
119883119862
|O| 119892
4119862
11520 11988611988721198861198873119886
11520 1198861198873119886119887119886119887311988611988721198861198872
5760 119887311988611988741198861198872119886
5760 11988611988731198861198871198861198872
2880 11988611988711988611988741198861198872119886
1440 119887411988611988741198861198873119886119887211988611988721198861198873119886
1440 1198861198872119886119887311988611988731198861198872119886119887119886119887
720 11988611988711988611988711988611988711988611988721198861198871198861198873119886
720 119886119887311988611988741198861198872119886
4119863
23040 1198871198861198873119886119887119886119887211988611988731198861198872
11520 119886119887119886119887311988611988731198861198871198861198873
5760 119886119887411988611988711988611988741198861198872119886
2880 11988611988711988611988711988611988741198861198872119886119887411988611988741198861198873119886
2880 1198861198872119886119887119886119887119886119887
311988611988711988611988741198861198874119886
4119864
23040 1198871198861198871198861198871198861198872
23040 119886119887311988611988721198861198872
11520 1198861198873119886119887211988611988741198861198872
11520 119887311988611988711988611988731198861198872119886
5760 11988731198861198874119886
5760 1198861198871198861198872
5760 1198874119886119887311988611988721198861198871198861198873119886
4119865
23040 119886119887119886119887211988611988721198861198872
23040 1198873119886119887311988611988731198861198874119886
11520 119887411988611988741198861198872119886119887411988611988731198861198872119886
11520 119886119887311988611988741198861198873119886119887211988611988721198861198872
5760 1198872119886119887411988611988721198861198872119886119887211988611988731198861198872119886
5760 1198872119886119887211988611988731198861198873119886119887411988611988721198861198873119886
5760 1198861198873119886119887211988611988741198861198872119886119887311988611988731198861198872119886
5760 1198861198874119886119887119886119887119886119887
3119886119887211988611988721198861198873
5760 119886119887211988611988731198861198871198861198872119886119887211988611988731198861198873
5760 1198874119886119887411988611988721198861198872119886119887211988611988731198861198872119886
2880 119887311988611988721198861198871198861198872
2880 1198873119886119887411988611988731198861198872
1920 11988611988711988611988741198861198872119886119887211988611988741198861198873
1920 1198871198861198873119886119887311988611988711988611988721198861198874
4119866
46080 1198873119886119887311988611988721198861198873119886
46080 1198861198872119886119887311988611988721198861198872
23040 1198861198871198861198872119886119887211988611988731198861198872119886
23040 11988611988731198861198872119886119887311988611988731198861198874119886
5119860 73728 119886119887211988611988731198861198871198861198872119886
5119861184320 119887
21198861198872119886
122880 119887211988611988731198861198872
Table 42 Continued
119883119862
|O| 119892
6119860 92160 1198871198861198873119886119887411988611988731198861198872119886
6119861 122880 119887211988611988741198861198872119886
6119862
46080 1198872119886119887311988611988731198861198872119886
46080 119887311988611988711988611988721198861198872
15360 1198874119886119887211988611988731198861198873119886
15360 119886119887211988611988721198861198873119886119887
6119863
92160 119887411988611988711988611988721198861198874119886
23040 11988741198861198873119886119887411988611988721198861198872
15360 11988611988721198861198872119886119887119886
15360 119886119887411988611988731198861198873119886
6119864
92160 1198871198861198871198861198872
92160 1198871198861198874119886
46080 1198873119886119887211988611988731198861198872119886
46080 11988711988611988721198861198873119886
46080 11988721198861198871198861198874119886
46080 11988611988711988611988741198861198873
15360 119886119887211988611988711988611988741198861198871198861198872119886
6119865
92160 1198861198873119886119887119886
92160 11988611988741198861198872119886
46080 119887311988611988721198861198871198861198872119886
46080 1198861198874119886119887411988611988731198861198873
46080 1198861198873119886119887411988611988731198861198872
46080 11988721198861198872119886119887119886119887119886
7119860 737280 11988741198861198872
811986092160 119887
2119886119887119886
92160 11988611988741198861198873
8119861
92160 119886119887411988611988721198861198872119886119887119886
92160 119886119887411988611988731198861198873119886119887119886
46080 11988611988711988611988731198861198872119886
46080 119886119887311988611988721198861198874119886
23040 119887119886119887311988611988731198861198871198861198873
23040 11988721198861198874119886119887211988611988721198861198874
8119862
92160 1198873119886119887311988611988731198861198874
92160 1198872
92160 1198873
92160 119887119886119887211988611988721198861198872
8119863
92160 119887311988611988731198861198872
92160 11988611988721198861198873119886119887311988611988741198861198872
92160 119887411988611988731198861198873119886
92160 11988711988611988721198861198871198861198873119886
92160 119887311988611988721198861198872
14 Algebra
Table 42 Continued
119883119862
|O| 119892
8119864
92160 1198861198873119886
92160 1198861198872119886
46080 119886119887311988611988711988611988741198861198872
46080 119887311988611988711988611988741198861198872119886
46080 119886119887211988611988721198861198871198861198873
46080 1198872119886119887119886119887119886
46080 1198872119886119887411988611988731198861198873119886
46080 1198872119886119887211988611988731198861198873119886
46080 1198861198872119886119887211988611988731198861198873
46080 119886119887411988611988741198861198873
23040 1198861198874119886119887211988611988721198861198871198861198873
23040 11988741198861198874119886119887211988611988741198861198873
8119865
184320 11988611988741198861198871198861198873119886
184320 11988611988721198861198874119886119887119886
92160 119886119887119886119887311988611988711988611988721198861198872
92160 11988731198861198874119886119887411988611988731198861198872119886
9119860737280 119886119887
41198861198871198861198872
245760 119887211988611988741198861198872
10119860 368640 119887211988611988731198861198872119886
10119861368640 119886119887
21198861198872
368640 11988731198861198873119886
10119862368640 119886119887
4
368640 11988611988711988611988731198861198871198861198873119886
11119860 1474560 11988721198861198873119886
12119860122880 119887119886119887
21198861198871198861198874119886
122880 119887411988611988711988611988741198861198873
12119861368640 119887
211988611988711988611988731198861198872
184320 1198861198874119886119887211988611988741198861198872
12119862368640 119886119887
2119886119887211988611988731198861198872
184320 119886119887411988611988711988611988721198861198872
12119863
184320 119887411988611988711988611988721198861198872
184320 11988611988711988611988711988611988731198861198873119886
92160 119886119887411988611988721198861198873119886
92160 11988611988721198861198873119886119887119886
12119864368640 119887
311988611988721198861198874
368640 11988711988611988731198861198872
12119865
184320 11988721198861198871198861198872119886
184320 11988611988731198861198874119886
184320 11988611988711988611988731198861198873
184320 1198861198871198861198872119886
92160 1198861198872119886119887119886
92160 11988611988741198861198873119886
12119866 368640 11988711988611988741198861198872119886
Table 42 Continued
119883119862
|O| 119892
12119867
184320 1198874119886119887311988611988731198861198872
184320 119886119887311988611988721198861198873119886
184320 11988611988711988611988711988611988741198861198872119886
184320 1198861198874119886119887311988611988741198861198872119886
184320 119887311988611988721198861198872119886119887
184320 119886119887211988611988731198861198872119886
14119860 737280 1198861198874119886
15119860737280 119887
4119886
737280 1198871198861198872119886
16119860
368640 11988611988731198861198872
368640 11988611988721198861198871198861198874
368640 11988731198861198872119886
368640 11988711988611988741198861198873119886
18119860 737280 11988731198861198873
20119860737280 119886119887
41198861198872
737280 1198873119886119887119886
20119861737280 119886119887
41198861198874119886
737280 119886119887119886119887119886
24119860737280 119886119887
2
737280 1198873119886
24119861737280 119887
4
737280 119887
28119860737280 119886119887119886119887
31198861198872
737280 119886119887411988611988721198861198872
30119860737280 119887
21198861198874
737280 1198871198861198873
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R H Bing ldquoA homeomorphism between the 3-sphere andthe sum of two solid horned spheresrdquo Annals of MathematicsSecond Series vol 56 pp 354ndash362 1952
[2] D Montgomery and L Zippin ldquoExamples of transformationgroupsrdquo Proceedings of the American Mathematical Society vol5 pp 460ndash465 1954
[3] J W Morgan and H Bass Eds The Smith Conjecture vol 112of Pure and Applied Mathematics Academic Press OrlandoFla USA 1984 Papers presented at the symposium held atColumbia University New York NY USA 1979
[4] C M Edwards and G T Ruttimann ldquoInvolutive and Peircegradings in JBWlowast-triplesrdquo Communications in Algebra vol 31no 6 pp 2819ndash2848 2003
Algebra 15
[5] J Gornicki and B E Rhoades ldquoA general fixed point theoremfor involutionsrdquo Indian Journal of Pure and Applied Mathemat-ics vol 27 no 1 pp 13ndash23 1996
[6] S Montgomery Fixed Rings of Finite Automorphism Groupsof Associative Rings vol 818 of Lecture Notes in MathematicsSpringer Berlin Germany 1980
[7] T A Springer ldquoThe classification of involutions of simplealgebraic groupsrdquo Journal of the Faculty of Science University ofTokyo Section IA Mathematics vol 34 no 3 pp 655ndash670 1987
[8] T A Springer ldquoSome results on algebraic group s with invo-lutionsrdquo in Algebraic Groups and Related Topics (KyotoNagoya1983) vol 6 of Advanced Studies in Pure Mathematics pp 525ndash543 North-Holland Amsterdam The Netherlands 1985
[9] W Feit and J GThompson ldquoSolvability of groups of odd orderrdquoPacific Journal of Mathematics vol 13 pp 775ndash1029 1963
[10] R Brauer and K A Fowler ldquoOn groups of even orderrdquo Annalsof Mathematics vol 62 no 3 pp 565ndash583 1955
[11] D Gorenstein The Classification of Finite Ssimple Groups Vol1 Groups of Noncharacteristic 2 Type The University Series inMathematics Plenum Press New York NY USA 1983
[12] D Gorenstein R Lyons and R Solomon The Classification ofthe Finite Simple Groups vol 40 of Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 1994
[13] C Bates and P Rowley ldquoInvolutions in Conwayrsquos largest simplegrouprdquo LMS Journal of Computation andMathematics vol 7 pp337ndash351 2004
[14] P Rowley and P Taylor ldquoPoint-line collinearity graphs of twosporadic minimal parabolic geometriesrdquo Journal of Algebra vol331 pp 304ndash310 2011
[15] P Rowley and P Taylor ldquoInvolutions in Jankorsquos simple group 1198694rdquoLMS Journal of Computation and Mathematics vol 14 pp 238ndash253 2011
[16] P Taylor ldquoInvolutions in Fischerrsquos sporadic groupsrdquo preprinthttpeprintsmamanacuk1622
[17] The GAP Group GAP-Groups Algorithms Programming Ver-sion 43 2002 httpwwwgap-systemorg
[18] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I The user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997
[19] R Wilson P Walsh J Tripp et al ldquoAtlas o f finite grouprepresentationsrdquo httpbrauermathsqmulacukAtlasv3
[20] J H Conway R T Curtis S P Norton R A Parker and R AWilson Atlas of Finite Groups Clarendon Oxford UK 1985
[21] R AWilson ldquoStandard generators for sporadic simple groupsrdquoJournal of Algebra vol 184 no 2 pp 505ndash515 1996
[22] D Gorenstein Finite Groups Chelsea Publishing New YorkNY USA 2nd edition 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Algebra 7
Table 22
119883119862
|O| 119892
2119860
168 1198872119886119887119886119887119886119887
2119886119887
14 119886119887119886
14 1198861198871198861198872119886119887
2119861 84 1198872119886119887
3119860 896 119887
3119861 128 119887211988611988721198861198871198861198872119886119887119886119887
Table 23
119883119862
|O| 119892
4119860112 119887119886119887119886119887119886119887
112 1198872119886119887211988611988721198861198872
4119861
1344 1198861198872119886119887119886
672 11988611988721198861198871198861198872119886119887
672 11988721198861198871198861198872119886119887119886
5119860 1792 119887119886119887119886
6119860 5376 119886
Table 24
119883119862
|O| 119892
211986060 119887
2119886119887119886119887211988611988721198861198871198861198872119886119887119886119887119886119887
2119886119887
15 11988711988611988721198861198872119886119887119886119887119886119887
2
2119861
120 119887119886119887211988611988711988611988721198861198871198861198872
80 119887211988611988711988611988721198861198871198861198872119886119887
2 (119887119886)8(1198872119886)21198871198861198872(119886119887)2(1198861198872)2(119886119887)3(1198861198872119886119887)2119887
3119860 320 119887211988611988721198861198871198861198872119886119887119886119887119886119887
3119861 960 11988721198861198871198861198872119886119887
4119860120 119887119886119887
21198861198872
120 1198871198861198871198861198872
4119861
480 1198872119886119887119886119887119886119887
2119886119887
240 1198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
240 1198872119886119887119886119887119886119887119886119887119886119887119886119887
4119862
960 1198871198861198872119886119887
960 11988721198861198871198861198872
160 1198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
21198861198872
160 1198872119886119887119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
5119860 1920 119887119886119887
6119860 1920 1198871198861198871198861198871198861198872
61198611920 119887119886119887
2
1920 1198872119886119887
101198601920 119887
21198861198871198861198872119886119887119886119887
1920 1198872119886119887119886119887119886119887119886119887
2
11119860 7680 119887119886119887119886119887
121198613840 119887
3840 1198872
3102 119883 = 2119861 119905 = 119886 See Table 24
311 119870 cong 119872119888119871
3111119883 = 2119860 119905 = 119886 See Table 25
3112119883 = 2119861 119905 = 119888 See Table 26
Table 25
119883119862
|O| 119892
2119860 210 119887211988611988731198861198872
3119861 2240 1198871198861198871198861198872
4119860 5040 1198872119886119887
5119861 8064 119887
6119861 6720 1198872
Table 26
119883119862
|O| 119892
2119860 165 11988611988721198861198871198861198872119886119887
3119860 220 1198872119886119887311988611988721198861198871198861198872119886
3119861660 119886119887
21198861198872
220 1198872119886119887119886119887119886119887
3
41198601980 119887
21198861198873119886119887119886119887119886
990 119886
5119860792 119887
211988611988721198861198873
792 1198872119886119887119886119887119886119887
2
5119861 7920 1198872
6119860 1980 119886119887
6119861
1980 1198871198861198872119886119887119886119887
1980 1198861198872119886119887
660
660
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198873
11988711988611988731198861198874119886119887
71198603960 119887119886119887119886119887
2119886
3960 1198871198861198872
71198613960 119887119886119887
3119886
3960 1198872119886
81198603960 119887
21198861198872119886
3960 11988611988731198861198873
9119860 2640 1198872119886119887119886
9119861 2640 1198861198871198861198872119886119887
101198603960 119887119886119887119886119887
3960 119886119887119886119887119886
121198603960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198873
1198871198861198873
141198603960 119886119887
21198861198873
3960 1198871198861198872119886119887
141198613960 119887119886119887119886119887119886
3960 1198872119886119887119886119887119886
151198603960 119887119886119887
3960 1198874
151198613960 119886119887119886119887
3960 119887
301198603960 119887119886
3960 1198871198861198872119886
301198613960 119887
41198861198872119886
3960 1198861198872
8 Algebra
Table 27
119883119862
|O| 119892
2119860240 119887119886119887119886119887119886119887
2 1198875(1198861198876)2(119886119887)31198875119886(11988721198861198873)2
2119861 315 119887211988611988711988611988721198861198872119886119887
3119860 1344 1198872119886119887
3119861 960 1198871198861198874119886119887
4119860480 119887
31198861198871198861198872119886119887
480 11988721198861198871198861198873119886119887
41198612520
2520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887
1198874119886119887
5119860 2688 1198873
6119860 13440 119887
Table 28
119883119862
|O| 119892
2119860 84 (119886119887)21198875(119886119887)2
2119861
168
168
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)2
(119886119887)1198874(119886119887)119887
2(119886119887)119887
2(119886119887)119887
4
14
14
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198874(119886119887)119887
4(119886119887)119887
2(119886119887)119887
2
1198872(119886119887)119887
2(119886119887)21198874(119886119887)119887
2(119886119887)119887
2
3119860 896 1198874(119886119887)119887
2
3119861 2688 1198872
4119861
1344
1344
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)119887
2(119886119887)119887
2
1198872(119886119887)119887
4(119886119887)21198874
112
112
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)1198872(119886119887)41198874(119886119887)2
(119886119887)31198874(119886119887)41198872
4119862
1344
1344
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)1198874(119886119887)119887
4
1198874(119886119887)119887
2(119886119887)119887
2(119886119887)31198872
672
672
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198872(119886119887)3
1198872(119886119887)41198872(119886119887)2
672
672
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)41198872(119886119887)2
(119886119887)4119887(119886119887)
2
5119860 5376 (119886119887)3
6119860 5376 (119886119887)1198872(119886119887)119887
2(119886119887)4
61198615376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198872(119886119887)
(119886119887)1198872(119886119887)
7119862 3072 1198872(119886119887)119887
2(119886119887)119887
2(119886119887)2
81198605376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)21198872(119886119887)21198872
(119886119887)1198872(119886119887)21198872(119886119887)
101198605376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)2
1198872(119886119887)119887
2(119886119887)119887
2
1211986110752
10752
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)2
1198874(119886119887)2
15119860 21504 1198872(119886119887)3
17119860 21504 (119886119887)1198872(119886119887)2
17119861 21504 (119886119887)
21119860 21504 (119886119887)1198872(119886119887)119887
2
21119861 21504 1198872(119886119887)
Table 29
119883119862
|O| 119892
2119860105 (119886119887)119887(119886119887)
3119887(119886119887)
3
63 119887611988611988731198861198873119886119887211988611988761198861198872119886119887611988611988761198861198874119886
2119861 315 (1198861198872)2(119887119886)21198872119886
3119860630 (119886119887)119887(119886119887)119887(119886119887)119887
126 119887(119886119887)21198876
3119861 2520 (119886119887)1198876
41198601260 (119886119887)
21198875(119886119887)
1260 (119886119887)119887(119886119887)1198873
4119861
945
945
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)21198872(119886119887)119887
119887(119886119887)119887(119886119887)1198872(119886119887)119887
315
315
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119886119887)5119887
119887(119886119887)31198872(119886119887)119887
41198621890
1890
1003816100381610038161003816100381610038161003816100381610038161003816
1198873(119886119887)2
(119886119887)1198872(119886119887)2119887
5119860 2520 (119886119887)3119887(119886119887)
6119860
3780 (119886119887)31198872(119886119887)2
1890 119887(119886119887)119887(119886119887)
1890 119887(119886119887)119887(119886119887)119887(119886119887)3
1260 119887(119886119887)1198872(119886119887)119887(119886119887)
2
6119861 7560 (119886119887)
71198601890 (119886119887)119887(119886119887)
41198872
90 119887(119886119887)1198873(119886119887)2119887
71198611890 119887
5(119886119887)119887(119886119887)
90 1198875(119886119887)119887
5(119886119887)119887
2(119886119887)119887
5
7119862 2160 119887(119886119887)119887(119886119887)4
71198637560 (119886119887)119887(119886119887)
4
1080 (119886119887)1198873(119886119887)
71198647560 (119886119887)
3119887(119886119887)
2
1080 (119886119887)1198874(119886119887)2119887(119886119887)
8119860
3780
3780
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)2119887(119886119887)
2
(119886119887)4
3780
3780
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)3119887(119886119887)119887(119886119887)119887
(119886119887)119887(119886119887)119887(119886119887)2
10119860 7560 (119886119887)3
121198607560 119887(119886119887)119887
7560 119887(119886119887)1198872
121198617560
7560
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119886119887)2119887
119887
14119860
3780 (119886119887)1198873(119886119887)2
3780 (119886119887)21198872(119886119887)
1890 (119886119887)1198872(119886119887)
1890 119887(119886119887)3119887
14119861
3780 (119886119887)119887(119886119887)2119887(119886119887)
3780 (119886119887)1198874
1890 (119886119887)21198872(119886119887)119887(119886119887)
2
1890 119887(119886119887)119887(119886119887)2
Algebra 9
Table 29 Continued
119883119862
|O| 119892
141198627560 119887(119886119887)
7560 119887(119886119887)3
141198637560 119887
3(119886119887)
7560 119887(119886119887)2
15119860 15120 (119886119887)1198872
21119860 15120 (119886119887)4119887(119886119887)
21119861 15120 (119886119887)119887
21119862 15120 (119886119887)3119887
21119863 15120 1198872
281198607560 119887
2(119886119887)119887
2
7560 (119886119887)2
281198617560 119887
4(119886119887)3
7560 (119886119887)119887(119886119887)2
Table 30
119883119862|O| 119892
2119860
640 11988611988731198861198872119886119887119886
480 11988611988721198861198872119886119887119886119887119886119887119886119887119886119887
21198861198872119886
240 11988611988721198861198873119886119887311988611988731198861198873119886119887211988611988711988611988721198861198873119886119887119886119887119886119887119886119887
31198861198873
30 1198861198872119886119887119886119887119886119887211988611988721198861198873119886119887311988611988711988611988711988611988721198861198873119886119887211988611988721198861198873119886119887211988611988711988611988711988611988731198863119860 10240 119886119887
3119886119887119886119887119886119887119886119887119886119887
2119886
4119860 5120 119886119887119886119887119886119887119886
4119862
7680 119886119887119886119887211988611988721198861198872119886119887119886
3840 119886119887211988611988731198861198871198861198872119886119887119886
3840 119886119887119886119887211988611988711988611988731198861198872119886
411986315360 119886119887119886
7680 1198861198872119886119887211988611988711988611988721198861198871198861198872119886
5119860 12288 11988611988731198861198871198861198872119886119887119886119887119886119887119886
5119861 4096 119886119887119886119887119886
611986061440 119886119887
21198861198871198861198872119886119887119886
30720 119886119887119886119887119886119887119886119887119886119887119886
7119860 61440 119886
811986261440 119886119887119886119887
21198861198872119886
61440 11988611988721198861198872119886119887119886
10119860 61440 11988611988721198861198872119886119887119886119887119886119887119886
12119860122880 1198861198872119886119887119886
13119860 61440 1198861198873119886119887119886
312 119870 cong 119867119890
3121119883 = 2119860 119905 = 119886 See Table 27
3122119883 = 2119861 119905 = (1198861198872)6 See Table 28
3123 119883 = 2119862 119905 = 1198893 See Table 29
313 119870 cong 119877119906
3131 119883 = 2119860 119905 = 1198872 See Table 30
3132 119883 = 2119861 119905 = 119886 See Table 31
314 119870 cong 119878119906119911
3141119883 = 2119860 119905 = ((119886119887)21198861198872)6 See Table 32
Table 31
119883119862 |O| 119892
2119860 455 119887119886119887311988611988721198861198873119886119887
2119861
455 11988711988611988731198861198872119886119887119886119887119886119887119886119887119886119887
455 11988711988611988711988611988711988611988711988611988711988611988721198861198873119886119887
1 1198873119886119887311988611988731198861198872119886119887119886119887119886119887
211988611988721198861198873119886119887119886119887211988611988721198861198872
1 1198871198861198872119886119887119886119887311988611988711988611988721198861198872119886119887119886119887211988611988711988611988731198861198873119886119887119886119887119886119887
31198861198871198861198872119886119887119886119887211988611988731198861198871198861198873
3119860 5824 1198872119886119887119886119887119886119887119886119887119886119887
4119860 3640 (119886119887)10119886(119886119887)15119886((119886119887)
5119886)2
4119862 3640 1198873119886119887119886119887119886119887
21198861198872
4119863 3640 119887311988611988711988611988721198861198873119886119887
51198605824 119887
311988611988711988611988721198861198871198861198872
5824 119887211988611988731198861198872119886119887
5119861 29120 1198872119886119887119886119887
6119860 29120 1198872119886119887119886119887119886119887
2119886119887
7119860
29120 11988721198861198872
29120 119887
29120 1198872
1011986029120 119887119886119887119886119887
2119886119887119886119887
29120 119887119886119887119886119887
1011986129120 119887119886119887119886119887
31198861198873119886119887
29120 11988711988611988731198861198873119886119887119886119887
12119860 58240 1198873119886119887119886119887119886119887
2119886119887
13119860
29120 119887119886119887
29120 119887119886119887119886119887119886119887
29120 11988731198861198872119886119887
1411986029120 119887
311988611988721198861198872119886119887119886119887
29120 11988711988611988721198861198872119886119887119886119887119886119887
1411986129120 119887
21198861198872119886119887119886119887119886119887
29120 11988711988611988711988611988711988611988721198861198872
1411986229120 119887119886119887
2119886119887119886119887119886119887119886119887
2
29120 1198872119886119887119886119887119886119887119886119887
2119886119887
15119860 116480 11988731198861198872119886119887119886119887119886119887
2011986058240 119887
31198861198871198861198872
58240 11988721198861198873119886119887
2611986029120 119887119886119887
2
29120 1198872119886119887
2611986129120 119887119886119887
21198861198872
29120 11988721198861198872119886119887
2611986229120 119887
3119886119887119886119887119886119887
29120 11988711988611988711988611988721198861198872
29119860 116480 1198871198861198872119886119887
29119861 116480 1198873119886119887119886119887
Table 32
119883119862
|O| 119892
2119860360 119886
54 ((119886119887)2119886)4119886((119886119887)
2119886)3
3119861 5120 11988721198861198872119886119887119886119887119886119887119886119887119886119887119886
3119862 9216 119887119886119887119886
4119860 1728 119887211988611988721198861198872119886119887119886119887119886119887
4119862 17280 1198872119886119887
5119861 55296 1198872119886119887119886119887119886
6119863 46080 119887
10 Algebra
Table 33119883119862|O| 119892
2119860 315 1198871198861198872119886119887119886119887119886119887
21198861198872119886119887211988611988721198861198872119886119887119886119887119886119887
2119886119887
2119861
630 1198872119886119887119886119887119886119887119886119887
2119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
21198861198872119886119887119886119887211988611988721198861198872119886119887
560 119887119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
2 (1198871198861198872119886)2(1198872119886)2(119887119886(1198872119886)2)2(1198872119886119887119886)2((1198872119886)21198872119886119887119886)21198871198861198872(119886119887)2
3119860 224 (1198871198861198872119886)3119887(119887119886)3(1198872119886)2(119887119886)5119887(119887119886)4(1198872119886)2119887
3119861 2240 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887
3119862 20160 11988711988611988721198861198871198861198872119886119887
41198602520 119887119886119887119886119887119886119887
2119886119887211988611988721198861198872119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886119887119886119887
4119861
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
2520 11988711988611988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
21198861198872119886119887
1260 119887211988611988721198861198871198861198872119886119887119886119887
4119862
5040 11988711988611988721198861198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
2520 11988711988611988711988611988711988611988721198861198872119886119887119886119887119886119887
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
4119863
10080 11988711988611988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
10080 11988711988611988711988611988721198861198872119886119887119886119887211988611988721198861198872119886119887
1120 (119886119887)4((119886119887)4119886)4119886(119886119887)8((119886119887)4119886)2
1120 (119886(1198872119886)4)2(1198872119886)8119886(119886(1198872119886)4)4(1198872119886)4
5119860 20160 119887119886119887119886119887
5119861 40320 1198872119886119887119886119887119886119887
2119886119887
6119860 10080 1198871198861198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
6119863 20160 11988711988611988721198861198872119886119887
611986440320 119887
2119886119887119886119887119886119887
2119886119887119886119887119886119887
40320 1198872119886119887119886119887211988611988721198861198872119886119887119886119887119886119887
7119860 80640 1198872119886119887119886119887119886119887119886119887
811986020160 119887119886119887
2119886119887119886119887211988611988721198861198872119886119887
20160 1198871198861198872119886119887211988611988721198861198871198861198872119886119887
8119861
20160 1198871198861198872119886119887119886119887119886119887119886119887
211988611988721198861198871198861198872119886119887
20160 11988711988611988711988611988721198861198872119886119887
20160 11988711988611988721198861198872119886119887119886119887
20160 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887119886119887
2119886119887
10080 11988721198861198872119886119887
10080 1198872119886119887119886119887
8119862
20160 11988721198861198871198861198872119886119887119886119887119886119887
20160 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
2119886119887
20160 1198872119886119887211988611988721198861198871198861198872119886119887
20160 1198871198861198872119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
10119860
40320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
119887211988611988711988611988721198861198871198861198872119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886119887
20160 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887
20160 11988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
1011986140320 119887119886119887119886119887119886119887119886119887
2119886119887
40320 1198871198861198872119886119887119886119887119886119887119886119887
11119860 161280 11988721198861198871198861198872119886119887119886119887
1211986040320 119887
2119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
20160 119887211988611988711988611988721198861198872119886119887119886119887119886119887
2119886119887
1211986140320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886119887
11988721198861198872119886119887211988611988721198861198872119886119887
12119862 40320 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
1211986380640 119887119886119887
2119886119887119886119887119886119887
80640 1198871198861198871198861198871198861198872119886119887
13119860 161280 119887119886119887
13119861 161280 11988721198861198872119886119887119886119887
14119860 80640 119887211988611988711988611988721198861198872119886119887119886119887
15119860 161280 119887
15119861 161280 1198872119886119887
15119862 161280 119887211988611988721198861198871198861198872119886119887119886119887119886119887
Table 33 Continued119883119862
|O| 119892
2011986080640
80640
1003816100381610038161003816100381610038161003816100381610038161003816
119887119886119887211988611988721198861198872119886119887119886119887
119887119886119887119886119887211988611988721198861198872119886119887
21119860 161280 119887119886119887211988611988721198861198872119886119887119886119887119886119887
21119861 161280 119887119886119887211988611988721198861198872119886119887
2411986080640 119887119886119887119886119887
2119886119887
80640 1198871198861198872119886119887119886119887
Table 34
119883119862
|O| 119892
2119860 315 ((119886119887)3119886)2119886((119886119887)
3119886)4119886(119886119887)
3((119886119887)3119886)2119886
2119861 1800 1198861198871198861198871198861198872
3119860 200 119887211988611988721198861198871198861198872119886119887119886119887
3119862 16800 11988611988721198861198872119886119887119886119887119886
41198603150 119887119886119887
21198861198871198861198872119886
630 1198871198861198872119886119887119886119887119886119887119886119887119886119887
2119886
4119862 12600 1198871198861198872119886119887119886
5119861 20160 119886119887119886
6119860 12600 1198861198871198861198871198861198872119886119887
6119864 100800 119887119886
7119860 100800 119887
811986025200 119886119887119886119887119886119887
25200 119887119886119887119886119887119886119887
12119860 50400 119886
Table 35
119883119862
|O| 119892
2119860 495 (119886119887)5119886 (119886119887)
2(119886 (119886119887)
5)4
(119886119887)5
2119861 792 11988721198861198871198861198872119886119887119886119887119886119887
2119886119887119886119887
3119860 264 119886(119886119887)9((119886119887)9119886)4119886(119886119887)
9((119886119887)9119886)2
31198615280 119886(119886119887)
6((119886119887)3119886)4
1760 (119886((119886119887)3119886)2)4119886
3119862 2640 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886
4119860
2970 11988711988611988711988611988721198861198871198861198872119886119887119886
495
495
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886
4119861 11880 119887119886119887211988611988721198861198871198861198872119886119887119886
4119862 11880 1198872119886119887
5119860 19008 119887119886119887119886
5119861 47520 1198861198872119886119887119886119887119886
6119860
11880 119886119887119886
3960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198871198861198872119886
11988611988711988611988721198861198872119886
611986115840 119886119887
2119886119887119886119887
15840 1198861198872119886119887119886119887119886119887119886119887119886119887
611986215840 119887
21198861198872119886119887119886
15840 1198872119886119887119886119887119886119887119886119887119886119887
21198861198872
611986347520 119886119887
2119886119887119886119887119886119887119886
15840 1198861198872119886119887119886119887119886119887
2119886119887119886
Algebra 11
Table 35 Continued
119883119862
|O| 119892
6119864 15840 119886119887119886119887119886119887119886119887211988611988711988611988721198861198872119886119887
7119860 47520 1198871198861198872119886119887119886
811986147520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886
119886119887119886119887119886119887
911986095040 119887119886119887119886119887119886119887
31680 119887119886119887119886119887211988611988721198861198871198861198872119886
911986195040 119887
2119886119887119886119887119886
31680 1198861198872119886119887119886119887119886119887119886119887119886119887119886
10119860 95040 119886119887
1011986147520 119887119886119887119886119887
21198861198872119886
47520 11988711988611988721198861198872119886119887119886
11119860 190080 119886
12119860 47520 1198872119886119887119886
12119861
47520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198872119886119887119886119887119886119887
2
119886119887119886119887119886119887119886119887119886
15840
15840
1003816100381610038161003816100381610038161003816100381610038161003816
11988721198861198872119886119887119886119887
119887119886119887119886119887119886
1211986247520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198872119886119887119886
1198861198871198861198871198861198871198861198871198861198871198861198872
12119864
47520 11988721198861198872119886
47520 119886119887119886119887
47520 119887119886119887119886119887119886119887119886119887
47520 1198872119886119887119886119887119886119887119886
1411986047520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198871198861198872119886119887119886119887
119887
15119862 190080 119886119887119886119887119886
1811986095040 119887119886119887
2119886119887119886119887119886
95040 11988611988721198861198872119886119887119886119887
1811986195040 119887
21198861198872119886119887119886119887119886
95040 1198871198861198871198861198871198861198872119886
2011986095040
95040
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198872119886119887119886
119887119886
Table 36
119883119862
|O| 119892
2119860 630 119886119887211988621198872119886
3119861 8960 11988611988721198862
3119862 1920 11988621198871198862
Table 37
119883119862
|O| 119892
4119861 30240 1198861198872119886
5119861 48384 1198861198871198862
6119862 80640 119886
Table 38
119883119862
|O| 119892
2119860 495 119887119886119887119886211988711988621198871198861198872119886119887
2119861396 119886119887
311988611988721198862119887119886
396 11988621198873119886119887211988621198871198862
31198601320 119887119886
211988711988621198872119886119887
440 11988611988721198861198871198861198872119886119887119886119887
31198612640 119886
211988721198861198871198861198873119886
1584 119886119887119886119887119886211988731198862
3119862 7920 11988711988611988721198861198872119886119887
4119860 2970 1198871198861198871198861198871198861198872119886119887119886
4119861
11880 1198862119887
5940 1198872
2970 1198871198861198871198861198872119886119887
51198609504 119887
21198861198871198862
9504 119887119886211988731198862119887119886
5119861 47520 11988621198872119886
611986011880 119887119886
211988721198861198872119886
3960 119886119887211988621198871198861198871198862119887119886
611986115840 119886
2
15840 119886
6119862
23760 1198871198861198871198861198873119886
23760 11988711988621198872119886119887119886
15840 1198871198861198873119886119887119886
611986423760 119887
2119886
23760 11988621198872
711986095040 119887119886
2
31680 1198871198861198872119886119887119886
811986023760 119887119886119887119886119887
2119886
23760 1198872119886119887119886119887119886119887
811986123760 119886119887119886119887
2119886
23760 11988721198862119887119886
3142 119883 = 2119861 119905 = 119886 See Table 33
3143 119883 = 2119862 119905 = 119888 See Table 34
3144 119883 = 2119863 119905 = (119888(119888119889)12)11 See Table 35
315 119870 cong 1198621199003
3151 119883 = 2119860 119905 = 1198872 See Tables 36 and 37
12 Algebra
Table 39
119883119862
|O| 119892
8119862
23760 11988611988721198861198872
23760 1198872119886211988721198862
23760 1198872119886119887119886
23760 11988711988621198871198861198873
911986031680 119887119886119887
21198861198873
31680 119886119887119886119887119886119887119886119887
9119861 63360 1198861198871198861198871198861198872
1011986047520 119886119887119886
21198872
47520 1198871198861198871198861198872
1011986147520 119886119887119886
47520 119886211988731198862
1211986023760 119886
21198871198862
23760 1198861198873119886
12119861
47520 119887
47520 1198861198871198861198871198862
47520 1198871198861198872119886119887
1411986095040 119887119886119887119886
95040 1198861198872119886119887119886
1511986095040 119886119887
2
95040 119886119887119886119887
15119861 190080 1198861198871198861198872
1811986095040 119886119887
2119886119887
95040 119887119886119887119886119887
21119860 190080 11988711988621198871198862
2411986095040 119886119887
2119886
95040 119886211988721198862
2411986195040 119886
2119887119886
95040 119887119886
3011986095040 119886119887119886
2
95040 11988611988731198862
3152 119883 = 2119861 119905 = ((1198862119887)3119886119887)5 See Tables 38 and 39
316 119870 cong 1198621199002
3161 119883 = 2119860 119905 = 119886 See Table 40
3162 119883 = 2119861 119905 = 119886(119886119887119886119887119886)2119886(119886119887119886119887119886)7 See Table 41
3163 119883 = 2119862 119905 = (119887minus1(1198861198872)2119886119887minus1(1198861198872)4)5 See Table 42
Table 40
119883119862
|O| 119892
2119861 1260 1198871198861198872119886119887
2119862 1008 11988731198861198871198861198873
3119861 14336 119887119886119887
4119862 40320 119887
Table 41
119883119862
|O| 119892
2119860 15 (1198874119886)2119887311988611988721198861198874(119887119886119887)
2119886(119887119886119887)
711988611988711988611988731198861198872(119886119887)2
21198611680 119887
31198861198871198861198873
210 11988741198861198874119886119887119886119887
2119862 2520 (119886119887)5119886(119886119887)
2119886(119886119887)
3
3119861 35840 1198872119886119887
4119860 1920 11988711988611988731198861198872119886119887119886119887
4119862 20160 1198871198861198872119886119887
4119864 161280 119887119886119887
411986513440 119887119886119887119886119887119886119887119886119887
13440 119887311988611988711988611988721198861198872
5119861 344064 1198872
6119864 430080 119887
Table 42
119883119862
|O| 119892
2119860240 119886119887
4119886119887211988611988711988611988741198861198873119886119887311988611988741198861198873119886119887119886
45 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887411988611988741198861198872119886
2119861
720 1198872119886
240 11988741198861198874119886119887311988611988741198861198874119886119887211988611988731198861198872119886
180 119886119887211988611988731198861198872119886119887119886119887311988611988731198861198872
30 1198861198871198861198874119886
2119862
1440 1198861198871198861198873119886119887311988611988721198861198871198861198872119886
1440 119886119887311988611988741198861198873119886119887211988611988721198861198874119886
1440 1198872119886119887119886119887119886119887
311988611988741198861198873
1152 11988611988711988611988731198861198873119886119887311988611988741198861198872119886
360 11988611988711988611988711988611988731198861198872119886119887119886119887311988611988731198861198872
3119860 10240 1198861198874119886119887211988611988711988611988721198861198871198861198872119886
3119861
15360 11988611988731198861198872119886
7680 11988731198861198872
1024 1198861198872119886119887211988611988731198861198873119886
41198603840 119887
411988611988741198861198873119886119887411988611988741198861198874
1440 1198861198871198861198872119886119887211988611988741198861198872
4119861
11520 1198871198861198874119886119887411988611988731198861198873119886
5760 119886119887211988611988711988611988721198861198871198861198873119886
5760 11988611988721198861198874119886119887311988611988741198861198873119886
1440 11988611988741198861198871198861198872119886119887411988611988721198861198871198861198873119886
1440 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887119886
720 119886119887119886119887411988611988721198861198872119886119887211988611988731198861198873
720 1198872119886119887411988611988721198861198872119886119887411988611988731198861198874119886
Algebra 13
Table 42 Continued
119883119862
|O| 119892
4119862
11520 11988611988721198861198873119886
11520 1198861198873119886119887119886119887311988611988721198861198872
5760 119887311988611988741198861198872119886
5760 11988611988731198861198871198861198872
2880 11988611988711988611988741198861198872119886
1440 119887411988611988741198861198873119886119887211988611988721198861198873119886
1440 1198861198872119886119887311988611988731198861198872119886119887119886119887
720 11988611988711988611988711988611988711988611988721198861198871198861198873119886
720 119886119887311988611988741198861198872119886
4119863
23040 1198871198861198873119886119887119886119887211988611988731198861198872
11520 119886119887119886119887311988611988731198861198871198861198873
5760 119886119887411988611988711988611988741198861198872119886
2880 11988611988711988611988711988611988741198861198872119886119887411988611988741198861198873119886
2880 1198861198872119886119887119886119887119886119887
311988611988711988611988741198861198874119886
4119864
23040 1198871198861198871198861198871198861198872
23040 119886119887311988611988721198861198872
11520 1198861198873119886119887211988611988741198861198872
11520 119887311988611988711988611988731198861198872119886
5760 11988731198861198874119886
5760 1198861198871198861198872
5760 1198874119886119887311988611988721198861198871198861198873119886
4119865
23040 119886119887119886119887211988611988721198861198872
23040 1198873119886119887311988611988731198861198874119886
11520 119887411988611988741198861198872119886119887411988611988731198861198872119886
11520 119886119887311988611988741198861198873119886119887211988611988721198861198872
5760 1198872119886119887411988611988721198861198872119886119887211988611988731198861198872119886
5760 1198872119886119887211988611988731198861198873119886119887411988611988721198861198873119886
5760 1198861198873119886119887211988611988741198861198872119886119887311988611988731198861198872119886
5760 1198861198874119886119887119886119887119886119887
3119886119887211988611988721198861198873
5760 119886119887211988611988731198861198871198861198872119886119887211988611988731198861198873
5760 1198874119886119887411988611988721198861198872119886119887211988611988731198861198872119886
2880 119887311988611988721198861198871198861198872
2880 1198873119886119887411988611988731198861198872
1920 11988611988711988611988741198861198872119886119887211988611988741198861198873
1920 1198871198861198873119886119887311988611988711988611988721198861198874
4119866
46080 1198873119886119887311988611988721198861198873119886
46080 1198861198872119886119887311988611988721198861198872
23040 1198861198871198861198872119886119887211988611988731198861198872119886
23040 11988611988731198861198872119886119887311988611988731198861198874119886
5119860 73728 119886119887211988611988731198861198871198861198872119886
5119861184320 119887
21198861198872119886
122880 119887211988611988731198861198872
Table 42 Continued
119883119862
|O| 119892
6119860 92160 1198871198861198873119886119887411988611988731198861198872119886
6119861 122880 119887211988611988741198861198872119886
6119862
46080 1198872119886119887311988611988731198861198872119886
46080 119887311988611988711988611988721198861198872
15360 1198874119886119887211988611988731198861198873119886
15360 119886119887211988611988721198861198873119886119887
6119863
92160 119887411988611988711988611988721198861198874119886
23040 11988741198861198873119886119887411988611988721198861198872
15360 11988611988721198861198872119886119887119886
15360 119886119887411988611988731198861198873119886
6119864
92160 1198871198861198871198861198872
92160 1198871198861198874119886
46080 1198873119886119887211988611988731198861198872119886
46080 11988711988611988721198861198873119886
46080 11988721198861198871198861198874119886
46080 11988611988711988611988741198861198873
15360 119886119887211988611988711988611988741198861198871198861198872119886
6119865
92160 1198861198873119886119887119886
92160 11988611988741198861198872119886
46080 119887311988611988721198861198871198861198872119886
46080 1198861198874119886119887411988611988731198861198873
46080 1198861198873119886119887411988611988731198861198872
46080 11988721198861198872119886119887119886119887119886
7119860 737280 11988741198861198872
811986092160 119887
2119886119887119886
92160 11988611988741198861198873
8119861
92160 119886119887411988611988721198861198872119886119887119886
92160 119886119887411988611988731198861198873119886119887119886
46080 11988611988711988611988731198861198872119886
46080 119886119887311988611988721198861198874119886
23040 119887119886119887311988611988731198861198871198861198873
23040 11988721198861198874119886119887211988611988721198861198874
8119862
92160 1198873119886119887311988611988731198861198874
92160 1198872
92160 1198873
92160 119887119886119887211988611988721198861198872
8119863
92160 119887311988611988731198861198872
92160 11988611988721198861198873119886119887311988611988741198861198872
92160 119887411988611988731198861198873119886
92160 11988711988611988721198861198871198861198873119886
92160 119887311988611988721198861198872
14 Algebra
Table 42 Continued
119883119862
|O| 119892
8119864
92160 1198861198873119886
92160 1198861198872119886
46080 119886119887311988611988711988611988741198861198872
46080 119887311988611988711988611988741198861198872119886
46080 119886119887211988611988721198861198871198861198873
46080 1198872119886119887119886119887119886
46080 1198872119886119887411988611988731198861198873119886
46080 1198872119886119887211988611988731198861198873119886
46080 1198861198872119886119887211988611988731198861198873
46080 119886119887411988611988741198861198873
23040 1198861198874119886119887211988611988721198861198871198861198873
23040 11988741198861198874119886119887211988611988741198861198873
8119865
184320 11988611988741198861198871198861198873119886
184320 11988611988721198861198874119886119887119886
92160 119886119887119886119887311988611988711988611988721198861198872
92160 11988731198861198874119886119887411988611988731198861198872119886
9119860737280 119886119887
41198861198871198861198872
245760 119887211988611988741198861198872
10119860 368640 119887211988611988731198861198872119886
10119861368640 119886119887
21198861198872
368640 11988731198861198873119886
10119862368640 119886119887
4
368640 11988611988711988611988731198861198871198861198873119886
11119860 1474560 11988721198861198873119886
12119860122880 119887119886119887
21198861198871198861198874119886
122880 119887411988611988711988611988741198861198873
12119861368640 119887
211988611988711988611988731198861198872
184320 1198861198874119886119887211988611988741198861198872
12119862368640 119886119887
2119886119887211988611988731198861198872
184320 119886119887411988611988711988611988721198861198872
12119863
184320 119887411988611988711988611988721198861198872
184320 11988611988711988611988711988611988731198861198873119886
92160 119886119887411988611988721198861198873119886
92160 11988611988721198861198873119886119887119886
12119864368640 119887
311988611988721198861198874
368640 11988711988611988731198861198872
12119865
184320 11988721198861198871198861198872119886
184320 11988611988731198861198874119886
184320 11988611988711988611988731198861198873
184320 1198861198871198861198872119886
92160 1198861198872119886119887119886
92160 11988611988741198861198873119886
12119866 368640 11988711988611988741198861198872119886
Table 42 Continued
119883119862
|O| 119892
12119867
184320 1198874119886119887311988611988731198861198872
184320 119886119887311988611988721198861198873119886
184320 11988611988711988611988711988611988741198861198872119886
184320 1198861198874119886119887311988611988741198861198872119886
184320 119887311988611988721198861198872119886119887
184320 119886119887211988611988731198861198872119886
14119860 737280 1198861198874119886
15119860737280 119887
4119886
737280 1198871198861198872119886
16119860
368640 11988611988731198861198872
368640 11988611988721198861198871198861198874
368640 11988731198861198872119886
368640 11988711988611988741198861198873119886
18119860 737280 11988731198861198873
20119860737280 119886119887
41198861198872
737280 1198873119886119887119886
20119861737280 119886119887
41198861198874119886
737280 119886119887119886119887119886
24119860737280 119886119887
2
737280 1198873119886
24119861737280 119887
4
737280 119887
28119860737280 119886119887119886119887
31198861198872
737280 119886119887411988611988721198861198872
30119860737280 119887
21198861198874
737280 1198871198861198873
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R H Bing ldquoA homeomorphism between the 3-sphere andthe sum of two solid horned spheresrdquo Annals of MathematicsSecond Series vol 56 pp 354ndash362 1952
[2] D Montgomery and L Zippin ldquoExamples of transformationgroupsrdquo Proceedings of the American Mathematical Society vol5 pp 460ndash465 1954
[3] J W Morgan and H Bass Eds The Smith Conjecture vol 112of Pure and Applied Mathematics Academic Press OrlandoFla USA 1984 Papers presented at the symposium held atColumbia University New York NY USA 1979
[4] C M Edwards and G T Ruttimann ldquoInvolutive and Peircegradings in JBWlowast-triplesrdquo Communications in Algebra vol 31no 6 pp 2819ndash2848 2003
Algebra 15
[5] J Gornicki and B E Rhoades ldquoA general fixed point theoremfor involutionsrdquo Indian Journal of Pure and Applied Mathemat-ics vol 27 no 1 pp 13ndash23 1996
[6] S Montgomery Fixed Rings of Finite Automorphism Groupsof Associative Rings vol 818 of Lecture Notes in MathematicsSpringer Berlin Germany 1980
[7] T A Springer ldquoThe classification of involutions of simplealgebraic groupsrdquo Journal of the Faculty of Science University ofTokyo Section IA Mathematics vol 34 no 3 pp 655ndash670 1987
[8] T A Springer ldquoSome results on algebraic group s with invo-lutionsrdquo in Algebraic Groups and Related Topics (KyotoNagoya1983) vol 6 of Advanced Studies in Pure Mathematics pp 525ndash543 North-Holland Amsterdam The Netherlands 1985
[9] W Feit and J GThompson ldquoSolvability of groups of odd orderrdquoPacific Journal of Mathematics vol 13 pp 775ndash1029 1963
[10] R Brauer and K A Fowler ldquoOn groups of even orderrdquo Annalsof Mathematics vol 62 no 3 pp 565ndash583 1955
[11] D Gorenstein The Classification of Finite Ssimple Groups Vol1 Groups of Noncharacteristic 2 Type The University Series inMathematics Plenum Press New York NY USA 1983
[12] D Gorenstein R Lyons and R Solomon The Classification ofthe Finite Simple Groups vol 40 of Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 1994
[13] C Bates and P Rowley ldquoInvolutions in Conwayrsquos largest simplegrouprdquo LMS Journal of Computation andMathematics vol 7 pp337ndash351 2004
[14] P Rowley and P Taylor ldquoPoint-line collinearity graphs of twosporadic minimal parabolic geometriesrdquo Journal of Algebra vol331 pp 304ndash310 2011
[15] P Rowley and P Taylor ldquoInvolutions in Jankorsquos simple group 1198694rdquoLMS Journal of Computation and Mathematics vol 14 pp 238ndash253 2011
[16] P Taylor ldquoInvolutions in Fischerrsquos sporadic groupsrdquo preprinthttpeprintsmamanacuk1622
[17] The GAP Group GAP-Groups Algorithms Programming Ver-sion 43 2002 httpwwwgap-systemorg
[18] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I The user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997
[19] R Wilson P Walsh J Tripp et al ldquoAtlas o f finite grouprepresentationsrdquo httpbrauermathsqmulacukAtlasv3
[20] J H Conway R T Curtis S P Norton R A Parker and R AWilson Atlas of Finite Groups Clarendon Oxford UK 1985
[21] R AWilson ldquoStandard generators for sporadic simple groupsrdquoJournal of Algebra vol 184 no 2 pp 505ndash515 1996
[22] D Gorenstein Finite Groups Chelsea Publishing New YorkNY USA 2nd edition 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Algebra
Table 27
119883119862
|O| 119892
2119860240 119887119886119887119886119887119886119887
2 1198875(1198861198876)2(119886119887)31198875119886(11988721198861198873)2
2119861 315 119887211988611988711988611988721198861198872119886119887
3119860 1344 1198872119886119887
3119861 960 1198871198861198874119886119887
4119860480 119887
31198861198871198861198872119886119887
480 11988721198861198871198861198873119886119887
41198612520
2520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887
1198874119886119887
5119860 2688 1198873
6119860 13440 119887
Table 28
119883119862
|O| 119892
2119860 84 (119886119887)21198875(119886119887)2
2119861
168
168
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)2
(119886119887)1198874(119886119887)119887
2(119886119887)119887
2(119886119887)119887
4
14
14
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198874(119886119887)119887
4(119886119887)119887
2(119886119887)119887
2
1198872(119886119887)119887
2(119886119887)21198874(119886119887)119887
2(119886119887)119887
2
3119860 896 1198874(119886119887)119887
2
3119861 2688 1198872
4119861
1344
1344
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)119887
2(119886119887)119887
2
1198872(119886119887)119887
4(119886119887)21198874
112
112
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)1198872(119886119887)41198874(119886119887)2
(119886119887)31198874(119886119887)41198872
4119862
1344
1344
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)1198874(119886119887)119887
4
1198874(119886119887)119887
2(119886119887)119887
2(119886119887)31198872
672
672
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198872(119886119887)3
1198872(119886119887)41198872(119886119887)2
672
672
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)41198872(119886119887)2
(119886119887)4119887(119886119887)
2
5119860 5376 (119886119887)3
6119860 5376 (119886119887)1198872(119886119887)119887
2(119886119887)4
61198615376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)21198872(119886119887)
(119886119887)1198872(119886119887)
7119862 3072 1198872(119886119887)119887
2(119886119887)119887
2(119886119887)2
81198605376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)21198872(119886119887)21198872
(119886119887)1198872(119886119887)21198872(119886119887)
101198605376
5376
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)2
1198872(119886119887)119887
2(119886119887)119887
2
1211986110752
10752
1003816100381610038161003816100381610038161003816100381610038161003816
1198872(119886119887)119887
2(119886119887)2
1198874(119886119887)2
15119860 21504 1198872(119886119887)3
17119860 21504 (119886119887)1198872(119886119887)2
17119861 21504 (119886119887)
21119860 21504 (119886119887)1198872(119886119887)119887
2
21119861 21504 1198872(119886119887)
Table 29
119883119862
|O| 119892
2119860105 (119886119887)119887(119886119887)
3119887(119886119887)
3
63 119887611988611988731198861198873119886119887211988611988761198861198872119886119887611988611988761198861198874119886
2119861 315 (1198861198872)2(119887119886)21198872119886
3119860630 (119886119887)119887(119886119887)119887(119886119887)119887
126 119887(119886119887)21198876
3119861 2520 (119886119887)1198876
41198601260 (119886119887)
21198875(119886119887)
1260 (119886119887)119887(119886119887)1198873
4119861
945
945
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)21198872(119886119887)119887
119887(119886119887)119887(119886119887)1198872(119886119887)119887
315
315
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119886119887)5119887
119887(119886119887)31198872(119886119887)119887
41198621890
1890
1003816100381610038161003816100381610038161003816100381610038161003816
1198873(119886119887)2
(119886119887)1198872(119886119887)2119887
5119860 2520 (119886119887)3119887(119886119887)
6119860
3780 (119886119887)31198872(119886119887)2
1890 119887(119886119887)119887(119886119887)
1890 119887(119886119887)119887(119886119887)119887(119886119887)3
1260 119887(119886119887)1198872(119886119887)119887(119886119887)
2
6119861 7560 (119886119887)
71198601890 (119886119887)119887(119886119887)
41198872
90 119887(119886119887)1198873(119886119887)2119887
71198611890 119887
5(119886119887)119887(119886119887)
90 1198875(119886119887)119887
5(119886119887)119887
2(119886119887)119887
5
7119862 2160 119887(119886119887)119887(119886119887)4
71198637560 (119886119887)119887(119886119887)
4
1080 (119886119887)1198873(119886119887)
71198647560 (119886119887)
3119887(119886119887)
2
1080 (119886119887)1198874(119886119887)2119887(119886119887)
8119860
3780
3780
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)2119887(119886119887)
2
(119886119887)4
3780
3780
1003816100381610038161003816100381610038161003816100381610038161003816
(119886119887)3119887(119886119887)119887(119886119887)119887
(119886119887)119887(119886119887)119887(119886119887)2
10119860 7560 (119886119887)3
121198607560 119887(119886119887)119887
7560 119887(119886119887)1198872
121198617560
7560
1003816100381610038161003816100381610038161003816100381610038161003816
119887(119886119887)2119887
119887
14119860
3780 (119886119887)1198873(119886119887)2
3780 (119886119887)21198872(119886119887)
1890 (119886119887)1198872(119886119887)
1890 119887(119886119887)3119887
14119861
3780 (119886119887)119887(119886119887)2119887(119886119887)
3780 (119886119887)1198874
1890 (119886119887)21198872(119886119887)119887(119886119887)
2
1890 119887(119886119887)119887(119886119887)2
Algebra 9
Table 29 Continued
119883119862
|O| 119892
141198627560 119887(119886119887)
7560 119887(119886119887)3
141198637560 119887
3(119886119887)
7560 119887(119886119887)2
15119860 15120 (119886119887)1198872
21119860 15120 (119886119887)4119887(119886119887)
21119861 15120 (119886119887)119887
21119862 15120 (119886119887)3119887
21119863 15120 1198872
281198607560 119887
2(119886119887)119887
2
7560 (119886119887)2
281198617560 119887
4(119886119887)3
7560 (119886119887)119887(119886119887)2
Table 30
119883119862|O| 119892
2119860
640 11988611988731198861198872119886119887119886
480 11988611988721198861198872119886119887119886119887119886119887119886119887119886119887
21198861198872119886
240 11988611988721198861198873119886119887311988611988731198861198873119886119887211988611988711988611988721198861198873119886119887119886119887119886119887119886119887
31198861198873
30 1198861198872119886119887119886119887119886119887211988611988721198861198873119886119887311988611988711988611988711988611988721198861198873119886119887211988611988721198861198873119886119887211988611988711988611988711988611988731198863119860 10240 119886119887
3119886119887119886119887119886119887119886119887119886119887
2119886
4119860 5120 119886119887119886119887119886119887119886
4119862
7680 119886119887119886119887211988611988721198861198872119886119887119886
3840 119886119887211988611988731198861198871198861198872119886119887119886
3840 119886119887119886119887211988611988711988611988731198861198872119886
411986315360 119886119887119886
7680 1198861198872119886119887211988611988711988611988721198861198871198861198872119886
5119860 12288 11988611988731198861198871198861198872119886119887119886119887119886119887119886
5119861 4096 119886119887119886119887119886
611986061440 119886119887
21198861198871198861198872119886119887119886
30720 119886119887119886119887119886119887119886119887119886119887119886
7119860 61440 119886
811986261440 119886119887119886119887
21198861198872119886
61440 11988611988721198861198872119886119887119886
10119860 61440 11988611988721198861198872119886119887119886119887119886119887119886
12119860122880 1198861198872119886119887119886
13119860 61440 1198861198873119886119887119886
312 119870 cong 119867119890
3121119883 = 2119860 119905 = 119886 See Table 27
3122119883 = 2119861 119905 = (1198861198872)6 See Table 28
3123 119883 = 2119862 119905 = 1198893 See Table 29
313 119870 cong 119877119906
3131 119883 = 2119860 119905 = 1198872 See Table 30
3132 119883 = 2119861 119905 = 119886 See Table 31
314 119870 cong 119878119906119911
3141119883 = 2119860 119905 = ((119886119887)21198861198872)6 See Table 32
Table 31
119883119862 |O| 119892
2119860 455 119887119886119887311988611988721198861198873119886119887
2119861
455 11988711988611988731198861198872119886119887119886119887119886119887119886119887119886119887
455 11988711988611988711988611988711988611988711988611988711988611988721198861198873119886119887
1 1198873119886119887311988611988731198861198872119886119887119886119887119886119887
211988611988721198861198873119886119887119886119887211988611988721198861198872
1 1198871198861198872119886119887119886119887311988611988711988611988721198861198872119886119887119886119887211988611988711988611988731198861198873119886119887119886119887119886119887
31198861198871198861198872119886119887119886119887211988611988731198861198871198861198873
3119860 5824 1198872119886119887119886119887119886119887119886119887119886119887
4119860 3640 (119886119887)10119886(119886119887)15119886((119886119887)
5119886)2
4119862 3640 1198873119886119887119886119887119886119887
21198861198872
4119863 3640 119887311988611988711988611988721198861198873119886119887
51198605824 119887
311988611988711988611988721198861198871198861198872
5824 119887211988611988731198861198872119886119887
5119861 29120 1198872119886119887119886119887
6119860 29120 1198872119886119887119886119887119886119887
2119886119887
7119860
29120 11988721198861198872
29120 119887
29120 1198872
1011986029120 119887119886119887119886119887
2119886119887119886119887
29120 119887119886119887119886119887
1011986129120 119887119886119887119886119887
31198861198873119886119887
29120 11988711988611988731198861198873119886119887119886119887
12119860 58240 1198873119886119887119886119887119886119887
2119886119887
13119860
29120 119887119886119887
29120 119887119886119887119886119887119886119887
29120 11988731198861198872119886119887
1411986029120 119887
311988611988721198861198872119886119887119886119887
29120 11988711988611988721198861198872119886119887119886119887119886119887
1411986129120 119887
21198861198872119886119887119886119887119886119887
29120 11988711988611988711988611988711988611988721198861198872
1411986229120 119887119886119887
2119886119887119886119887119886119887119886119887
2
29120 1198872119886119887119886119887119886119887119886119887
2119886119887
15119860 116480 11988731198861198872119886119887119886119887119886119887
2011986058240 119887
31198861198871198861198872
58240 11988721198861198873119886119887
2611986029120 119887119886119887
2
29120 1198872119886119887
2611986129120 119887119886119887
21198861198872
29120 11988721198861198872119886119887
2611986229120 119887
3119886119887119886119887119886119887
29120 11988711988611988711988611988721198861198872
29119860 116480 1198871198861198872119886119887
29119861 116480 1198873119886119887119886119887
Table 32
119883119862
|O| 119892
2119860360 119886
54 ((119886119887)2119886)4119886((119886119887)
2119886)3
3119861 5120 11988721198861198872119886119887119886119887119886119887119886119887119886119887119886
3119862 9216 119887119886119887119886
4119860 1728 119887211988611988721198861198872119886119887119886119887119886119887
4119862 17280 1198872119886119887
5119861 55296 1198872119886119887119886119887119886
6119863 46080 119887
10 Algebra
Table 33119883119862|O| 119892
2119860 315 1198871198861198872119886119887119886119887119886119887
21198861198872119886119887211988611988721198861198872119886119887119886119887119886119887
2119886119887
2119861
630 1198872119886119887119886119887119886119887119886119887
2119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
21198861198872119886119887119886119887211988611988721198861198872119886119887
560 119887119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
2 (1198871198861198872119886)2(1198872119886)2(119887119886(1198872119886)2)2(1198872119886119887119886)2((1198872119886)21198872119886119887119886)21198871198861198872(119886119887)2
3119860 224 (1198871198861198872119886)3119887(119887119886)3(1198872119886)2(119887119886)5119887(119887119886)4(1198872119886)2119887
3119861 2240 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887
3119862 20160 11988711988611988721198861198871198861198872119886119887
41198602520 119887119886119887119886119887119886119887
2119886119887211988611988721198861198872119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886119887119886119887
4119861
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
2520 11988711988611988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
21198861198872119886119887
1260 119887211988611988721198861198871198861198872119886119887119886119887
4119862
5040 11988711988611988721198861198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
2520 11988711988611988711988611988711988611988721198861198872119886119887119886119887119886119887
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
4119863
10080 11988711988611988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
10080 11988711988611988711988611988721198861198872119886119887119886119887211988611988721198861198872119886119887
1120 (119886119887)4((119886119887)4119886)4119886(119886119887)8((119886119887)4119886)2
1120 (119886(1198872119886)4)2(1198872119886)8119886(119886(1198872119886)4)4(1198872119886)4
5119860 20160 119887119886119887119886119887
5119861 40320 1198872119886119887119886119887119886119887
2119886119887
6119860 10080 1198871198861198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
6119863 20160 11988711988611988721198861198872119886119887
611986440320 119887
2119886119887119886119887119886119887
2119886119887119886119887119886119887
40320 1198872119886119887119886119887211988611988721198861198872119886119887119886119887119886119887
7119860 80640 1198872119886119887119886119887119886119887119886119887
811986020160 119887119886119887
2119886119887119886119887211988611988721198861198872119886119887
20160 1198871198861198872119886119887211988611988721198861198871198861198872119886119887
8119861
20160 1198871198861198872119886119887119886119887119886119887119886119887
211988611988721198861198871198861198872119886119887
20160 11988711988611988711988611988721198861198872119886119887
20160 11988711988611988721198861198872119886119887119886119887
20160 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887119886119887
2119886119887
10080 11988721198861198872119886119887
10080 1198872119886119887119886119887
8119862
20160 11988721198861198871198861198872119886119887119886119887119886119887
20160 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
2119886119887
20160 1198872119886119887211988611988721198861198871198861198872119886119887
20160 1198871198861198872119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
10119860
40320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
119887211988611988711988611988721198861198871198861198872119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886119887
20160 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887
20160 11988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
1011986140320 119887119886119887119886119887119886119887119886119887
2119886119887
40320 1198871198861198872119886119887119886119887119886119887119886119887
11119860 161280 11988721198861198871198861198872119886119887119886119887
1211986040320 119887
2119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
20160 119887211988611988711988611988721198861198872119886119887119886119887119886119887
2119886119887
1211986140320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886119887
11988721198861198872119886119887211988611988721198861198872119886119887
12119862 40320 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
1211986380640 119887119886119887
2119886119887119886119887119886119887
80640 1198871198861198871198861198871198861198872119886119887
13119860 161280 119887119886119887
13119861 161280 11988721198861198872119886119887119886119887
14119860 80640 119887211988611988711988611988721198861198872119886119887119886119887
15119860 161280 119887
15119861 161280 1198872119886119887
15119862 161280 119887211988611988721198861198871198861198872119886119887119886119887119886119887
Table 33 Continued119883119862
|O| 119892
2011986080640
80640
1003816100381610038161003816100381610038161003816100381610038161003816
119887119886119887211988611988721198861198872119886119887119886119887
119887119886119887119886119887211988611988721198861198872119886119887
21119860 161280 119887119886119887211988611988721198861198872119886119887119886119887119886119887
21119861 161280 119887119886119887211988611988721198861198872119886119887
2411986080640 119887119886119887119886119887
2119886119887
80640 1198871198861198872119886119887119886119887
Table 34
119883119862
|O| 119892
2119860 315 ((119886119887)3119886)2119886((119886119887)
3119886)4119886(119886119887)
3((119886119887)3119886)2119886
2119861 1800 1198861198871198861198871198861198872
3119860 200 119887211988611988721198861198871198861198872119886119887119886119887
3119862 16800 11988611988721198861198872119886119887119886119887119886
41198603150 119887119886119887
21198861198871198861198872119886
630 1198871198861198872119886119887119886119887119886119887119886119887119886119887
2119886
4119862 12600 1198871198861198872119886119887119886
5119861 20160 119886119887119886
6119860 12600 1198861198871198861198871198861198872119886119887
6119864 100800 119887119886
7119860 100800 119887
811986025200 119886119887119886119887119886119887
25200 119887119886119887119886119887119886119887
12119860 50400 119886
Table 35
119883119862
|O| 119892
2119860 495 (119886119887)5119886 (119886119887)
2(119886 (119886119887)
5)4
(119886119887)5
2119861 792 11988721198861198871198861198872119886119887119886119887119886119887
2119886119887119886119887
3119860 264 119886(119886119887)9((119886119887)9119886)4119886(119886119887)
9((119886119887)9119886)2
31198615280 119886(119886119887)
6((119886119887)3119886)4
1760 (119886((119886119887)3119886)2)4119886
3119862 2640 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886
4119860
2970 11988711988611988711988611988721198861198871198861198872119886119887119886
495
495
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886
4119861 11880 119887119886119887211988611988721198861198871198861198872119886119887119886
4119862 11880 1198872119886119887
5119860 19008 119887119886119887119886
5119861 47520 1198861198872119886119887119886119887119886
6119860
11880 119886119887119886
3960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198871198861198872119886
11988611988711988611988721198861198872119886
611986115840 119886119887
2119886119887119886119887
15840 1198861198872119886119887119886119887119886119887119886119887119886119887
611986215840 119887
21198861198872119886119887119886
15840 1198872119886119887119886119887119886119887119886119887119886119887
21198861198872
611986347520 119886119887
2119886119887119886119887119886119887119886
15840 1198861198872119886119887119886119887119886119887
2119886119887119886
Algebra 11
Table 35 Continued
119883119862
|O| 119892
6119864 15840 119886119887119886119887119886119887119886119887211988611988711988611988721198861198872119886119887
7119860 47520 1198871198861198872119886119887119886
811986147520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886
119886119887119886119887119886119887
911986095040 119887119886119887119886119887119886119887
31680 119887119886119887119886119887211988611988721198861198871198861198872119886
911986195040 119887
2119886119887119886119887119886
31680 1198861198872119886119887119886119887119886119887119886119887119886119887119886
10119860 95040 119886119887
1011986147520 119887119886119887119886119887
21198861198872119886
47520 11988711988611988721198861198872119886119887119886
11119860 190080 119886
12119860 47520 1198872119886119887119886
12119861
47520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198872119886119887119886119887119886119887
2
119886119887119886119887119886119887119886119887119886
15840
15840
1003816100381610038161003816100381610038161003816100381610038161003816
11988721198861198872119886119887119886119887
119887119886119887119886119887119886
1211986247520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198872119886119887119886
1198861198871198861198871198861198871198861198871198861198871198861198872
12119864
47520 11988721198861198872119886
47520 119886119887119886119887
47520 119887119886119887119886119887119886119887119886119887
47520 1198872119886119887119886119887119886119887119886
1411986047520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198871198861198872119886119887119886119887
119887
15119862 190080 119886119887119886119887119886
1811986095040 119887119886119887
2119886119887119886119887119886
95040 11988611988721198861198872119886119887119886119887
1811986195040 119887
21198861198872119886119887119886119887119886
95040 1198871198861198871198861198871198861198872119886
2011986095040
95040
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198872119886119887119886
119887119886
Table 36
119883119862
|O| 119892
2119860 630 119886119887211988621198872119886
3119861 8960 11988611988721198862
3119862 1920 11988621198871198862
Table 37
119883119862
|O| 119892
4119861 30240 1198861198872119886
5119861 48384 1198861198871198862
6119862 80640 119886
Table 38
119883119862
|O| 119892
2119860 495 119887119886119887119886211988711988621198871198861198872119886119887
2119861396 119886119887
311988611988721198862119887119886
396 11988621198873119886119887211988621198871198862
31198601320 119887119886
211988711988621198872119886119887
440 11988611988721198861198871198861198872119886119887119886119887
31198612640 119886
211988721198861198871198861198873119886
1584 119886119887119886119887119886211988731198862
3119862 7920 11988711988611988721198861198872119886119887
4119860 2970 1198871198861198871198861198871198861198872119886119887119886
4119861
11880 1198862119887
5940 1198872
2970 1198871198861198871198861198872119886119887
51198609504 119887
21198861198871198862
9504 119887119886211988731198862119887119886
5119861 47520 11988621198872119886
611986011880 119887119886
211988721198861198872119886
3960 119886119887211988621198871198861198871198862119887119886
611986115840 119886
2
15840 119886
6119862
23760 1198871198861198871198861198873119886
23760 11988711988621198872119886119887119886
15840 1198871198861198873119886119887119886
611986423760 119887
2119886
23760 11988621198872
711986095040 119887119886
2
31680 1198871198861198872119886119887119886
811986023760 119887119886119887119886119887
2119886
23760 1198872119886119887119886119887119886119887
811986123760 119886119887119886119887
2119886
23760 11988721198862119887119886
3142 119883 = 2119861 119905 = 119886 See Table 33
3143 119883 = 2119862 119905 = 119888 See Table 34
3144 119883 = 2119863 119905 = (119888(119888119889)12)11 See Table 35
315 119870 cong 1198621199003
3151 119883 = 2119860 119905 = 1198872 See Tables 36 and 37
12 Algebra
Table 39
119883119862
|O| 119892
8119862
23760 11988611988721198861198872
23760 1198872119886211988721198862
23760 1198872119886119887119886
23760 11988711988621198871198861198873
911986031680 119887119886119887
21198861198873
31680 119886119887119886119887119886119887119886119887
9119861 63360 1198861198871198861198871198861198872
1011986047520 119886119887119886
21198872
47520 1198871198861198871198861198872
1011986147520 119886119887119886
47520 119886211988731198862
1211986023760 119886
21198871198862
23760 1198861198873119886
12119861
47520 119887
47520 1198861198871198861198871198862
47520 1198871198861198872119886119887
1411986095040 119887119886119887119886
95040 1198861198872119886119887119886
1511986095040 119886119887
2
95040 119886119887119886119887
15119861 190080 1198861198871198861198872
1811986095040 119886119887
2119886119887
95040 119887119886119887119886119887
21119860 190080 11988711988621198871198862
2411986095040 119886119887
2119886
95040 119886211988721198862
2411986195040 119886
2119887119886
95040 119887119886
3011986095040 119886119887119886
2
95040 11988611988731198862
3152 119883 = 2119861 119905 = ((1198862119887)3119886119887)5 See Tables 38 and 39
316 119870 cong 1198621199002
3161 119883 = 2119860 119905 = 119886 See Table 40
3162 119883 = 2119861 119905 = 119886(119886119887119886119887119886)2119886(119886119887119886119887119886)7 See Table 41
3163 119883 = 2119862 119905 = (119887minus1(1198861198872)2119886119887minus1(1198861198872)4)5 See Table 42
Table 40
119883119862
|O| 119892
2119861 1260 1198871198861198872119886119887
2119862 1008 11988731198861198871198861198873
3119861 14336 119887119886119887
4119862 40320 119887
Table 41
119883119862
|O| 119892
2119860 15 (1198874119886)2119887311988611988721198861198874(119887119886119887)
2119886(119887119886119887)
711988611988711988611988731198861198872(119886119887)2
21198611680 119887
31198861198871198861198873
210 11988741198861198874119886119887119886119887
2119862 2520 (119886119887)5119886(119886119887)
2119886(119886119887)
3
3119861 35840 1198872119886119887
4119860 1920 11988711988611988731198861198872119886119887119886119887
4119862 20160 1198871198861198872119886119887
4119864 161280 119887119886119887
411986513440 119887119886119887119886119887119886119887119886119887
13440 119887311988611988711988611988721198861198872
5119861 344064 1198872
6119864 430080 119887
Table 42
119883119862
|O| 119892
2119860240 119886119887
4119886119887211988611988711988611988741198861198873119886119887311988611988741198861198873119886119887119886
45 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887411988611988741198861198872119886
2119861
720 1198872119886
240 11988741198861198874119886119887311988611988741198861198874119886119887211988611988731198861198872119886
180 119886119887211988611988731198861198872119886119887119886119887311988611988731198861198872
30 1198861198871198861198874119886
2119862
1440 1198861198871198861198873119886119887311988611988721198861198871198861198872119886
1440 119886119887311988611988741198861198873119886119887211988611988721198861198874119886
1440 1198872119886119887119886119887119886119887
311988611988741198861198873
1152 11988611988711988611988731198861198873119886119887311988611988741198861198872119886
360 11988611988711988611988711988611988731198861198872119886119887119886119887311988611988731198861198872
3119860 10240 1198861198874119886119887211988611988711988611988721198861198871198861198872119886
3119861
15360 11988611988731198861198872119886
7680 11988731198861198872
1024 1198861198872119886119887211988611988731198861198873119886
41198603840 119887
411988611988741198861198873119886119887411988611988741198861198874
1440 1198861198871198861198872119886119887211988611988741198861198872
4119861
11520 1198871198861198874119886119887411988611988731198861198873119886
5760 119886119887211988611988711988611988721198861198871198861198873119886
5760 11988611988721198861198874119886119887311988611988741198861198873119886
1440 11988611988741198861198871198861198872119886119887411988611988721198861198871198861198873119886
1440 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887119886
720 119886119887119886119887411988611988721198861198872119886119887211988611988731198861198873
720 1198872119886119887411988611988721198861198872119886119887411988611988731198861198874119886
Algebra 13
Table 42 Continued
119883119862
|O| 119892
4119862
11520 11988611988721198861198873119886
11520 1198861198873119886119887119886119887311988611988721198861198872
5760 119887311988611988741198861198872119886
5760 11988611988731198861198871198861198872
2880 11988611988711988611988741198861198872119886
1440 119887411988611988741198861198873119886119887211988611988721198861198873119886
1440 1198861198872119886119887311988611988731198861198872119886119887119886119887
720 11988611988711988611988711988611988711988611988721198861198871198861198873119886
720 119886119887311988611988741198861198872119886
4119863
23040 1198871198861198873119886119887119886119887211988611988731198861198872
11520 119886119887119886119887311988611988731198861198871198861198873
5760 119886119887411988611988711988611988741198861198872119886
2880 11988611988711988611988711988611988741198861198872119886119887411988611988741198861198873119886
2880 1198861198872119886119887119886119887119886119887
311988611988711988611988741198861198874119886
4119864
23040 1198871198861198871198861198871198861198872
23040 119886119887311988611988721198861198872
11520 1198861198873119886119887211988611988741198861198872
11520 119887311988611988711988611988731198861198872119886
5760 11988731198861198874119886
5760 1198861198871198861198872
5760 1198874119886119887311988611988721198861198871198861198873119886
4119865
23040 119886119887119886119887211988611988721198861198872
23040 1198873119886119887311988611988731198861198874119886
11520 119887411988611988741198861198872119886119887411988611988731198861198872119886
11520 119886119887311988611988741198861198873119886119887211988611988721198861198872
5760 1198872119886119887411988611988721198861198872119886119887211988611988731198861198872119886
5760 1198872119886119887211988611988731198861198873119886119887411988611988721198861198873119886
5760 1198861198873119886119887211988611988741198861198872119886119887311988611988731198861198872119886
5760 1198861198874119886119887119886119887119886119887
3119886119887211988611988721198861198873
5760 119886119887211988611988731198861198871198861198872119886119887211988611988731198861198873
5760 1198874119886119887411988611988721198861198872119886119887211988611988731198861198872119886
2880 119887311988611988721198861198871198861198872
2880 1198873119886119887411988611988731198861198872
1920 11988611988711988611988741198861198872119886119887211988611988741198861198873
1920 1198871198861198873119886119887311988611988711988611988721198861198874
4119866
46080 1198873119886119887311988611988721198861198873119886
46080 1198861198872119886119887311988611988721198861198872
23040 1198861198871198861198872119886119887211988611988731198861198872119886
23040 11988611988731198861198872119886119887311988611988731198861198874119886
5119860 73728 119886119887211988611988731198861198871198861198872119886
5119861184320 119887
21198861198872119886
122880 119887211988611988731198861198872
Table 42 Continued
119883119862
|O| 119892
6119860 92160 1198871198861198873119886119887411988611988731198861198872119886
6119861 122880 119887211988611988741198861198872119886
6119862
46080 1198872119886119887311988611988731198861198872119886
46080 119887311988611988711988611988721198861198872
15360 1198874119886119887211988611988731198861198873119886
15360 119886119887211988611988721198861198873119886119887
6119863
92160 119887411988611988711988611988721198861198874119886
23040 11988741198861198873119886119887411988611988721198861198872
15360 11988611988721198861198872119886119887119886
15360 119886119887411988611988731198861198873119886
6119864
92160 1198871198861198871198861198872
92160 1198871198861198874119886
46080 1198873119886119887211988611988731198861198872119886
46080 11988711988611988721198861198873119886
46080 11988721198861198871198861198874119886
46080 11988611988711988611988741198861198873
15360 119886119887211988611988711988611988741198861198871198861198872119886
6119865
92160 1198861198873119886119887119886
92160 11988611988741198861198872119886
46080 119887311988611988721198861198871198861198872119886
46080 1198861198874119886119887411988611988731198861198873
46080 1198861198873119886119887411988611988731198861198872
46080 11988721198861198872119886119887119886119887119886
7119860 737280 11988741198861198872
811986092160 119887
2119886119887119886
92160 11988611988741198861198873
8119861
92160 119886119887411988611988721198861198872119886119887119886
92160 119886119887411988611988731198861198873119886119887119886
46080 11988611988711988611988731198861198872119886
46080 119886119887311988611988721198861198874119886
23040 119887119886119887311988611988731198861198871198861198873
23040 11988721198861198874119886119887211988611988721198861198874
8119862
92160 1198873119886119887311988611988731198861198874
92160 1198872
92160 1198873
92160 119887119886119887211988611988721198861198872
8119863
92160 119887311988611988731198861198872
92160 11988611988721198861198873119886119887311988611988741198861198872
92160 119887411988611988731198861198873119886
92160 11988711988611988721198861198871198861198873119886
92160 119887311988611988721198861198872
14 Algebra
Table 42 Continued
119883119862
|O| 119892
8119864
92160 1198861198873119886
92160 1198861198872119886
46080 119886119887311988611988711988611988741198861198872
46080 119887311988611988711988611988741198861198872119886
46080 119886119887211988611988721198861198871198861198873
46080 1198872119886119887119886119887119886
46080 1198872119886119887411988611988731198861198873119886
46080 1198872119886119887211988611988731198861198873119886
46080 1198861198872119886119887211988611988731198861198873
46080 119886119887411988611988741198861198873
23040 1198861198874119886119887211988611988721198861198871198861198873
23040 11988741198861198874119886119887211988611988741198861198873
8119865
184320 11988611988741198861198871198861198873119886
184320 11988611988721198861198874119886119887119886
92160 119886119887119886119887311988611988711988611988721198861198872
92160 11988731198861198874119886119887411988611988731198861198872119886
9119860737280 119886119887
41198861198871198861198872
245760 119887211988611988741198861198872
10119860 368640 119887211988611988731198861198872119886
10119861368640 119886119887
21198861198872
368640 11988731198861198873119886
10119862368640 119886119887
4
368640 11988611988711988611988731198861198871198861198873119886
11119860 1474560 11988721198861198873119886
12119860122880 119887119886119887
21198861198871198861198874119886
122880 119887411988611988711988611988741198861198873
12119861368640 119887
211988611988711988611988731198861198872
184320 1198861198874119886119887211988611988741198861198872
12119862368640 119886119887
2119886119887211988611988731198861198872
184320 119886119887411988611988711988611988721198861198872
12119863
184320 119887411988611988711988611988721198861198872
184320 11988611988711988611988711988611988731198861198873119886
92160 119886119887411988611988721198861198873119886
92160 11988611988721198861198873119886119887119886
12119864368640 119887
311988611988721198861198874
368640 11988711988611988731198861198872
12119865
184320 11988721198861198871198861198872119886
184320 11988611988731198861198874119886
184320 11988611988711988611988731198861198873
184320 1198861198871198861198872119886
92160 1198861198872119886119887119886
92160 11988611988741198861198873119886
12119866 368640 11988711988611988741198861198872119886
Table 42 Continued
119883119862
|O| 119892
12119867
184320 1198874119886119887311988611988731198861198872
184320 119886119887311988611988721198861198873119886
184320 11988611988711988611988711988611988741198861198872119886
184320 1198861198874119886119887311988611988741198861198872119886
184320 119887311988611988721198861198872119886119887
184320 119886119887211988611988731198861198872119886
14119860 737280 1198861198874119886
15119860737280 119887
4119886
737280 1198871198861198872119886
16119860
368640 11988611988731198861198872
368640 11988611988721198861198871198861198874
368640 11988731198861198872119886
368640 11988711988611988741198861198873119886
18119860 737280 11988731198861198873
20119860737280 119886119887
41198861198872
737280 1198873119886119887119886
20119861737280 119886119887
41198861198874119886
737280 119886119887119886119887119886
24119860737280 119886119887
2
737280 1198873119886
24119861737280 119887
4
737280 119887
28119860737280 119886119887119886119887
31198861198872
737280 119886119887411988611988721198861198872
30119860737280 119887
21198861198874
737280 1198871198861198873
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R H Bing ldquoA homeomorphism between the 3-sphere andthe sum of two solid horned spheresrdquo Annals of MathematicsSecond Series vol 56 pp 354ndash362 1952
[2] D Montgomery and L Zippin ldquoExamples of transformationgroupsrdquo Proceedings of the American Mathematical Society vol5 pp 460ndash465 1954
[3] J W Morgan and H Bass Eds The Smith Conjecture vol 112of Pure and Applied Mathematics Academic Press OrlandoFla USA 1984 Papers presented at the symposium held atColumbia University New York NY USA 1979
[4] C M Edwards and G T Ruttimann ldquoInvolutive and Peircegradings in JBWlowast-triplesrdquo Communications in Algebra vol 31no 6 pp 2819ndash2848 2003
Algebra 15
[5] J Gornicki and B E Rhoades ldquoA general fixed point theoremfor involutionsrdquo Indian Journal of Pure and Applied Mathemat-ics vol 27 no 1 pp 13ndash23 1996
[6] S Montgomery Fixed Rings of Finite Automorphism Groupsof Associative Rings vol 818 of Lecture Notes in MathematicsSpringer Berlin Germany 1980
[7] T A Springer ldquoThe classification of involutions of simplealgebraic groupsrdquo Journal of the Faculty of Science University ofTokyo Section IA Mathematics vol 34 no 3 pp 655ndash670 1987
[8] T A Springer ldquoSome results on algebraic group s with invo-lutionsrdquo in Algebraic Groups and Related Topics (KyotoNagoya1983) vol 6 of Advanced Studies in Pure Mathematics pp 525ndash543 North-Holland Amsterdam The Netherlands 1985
[9] W Feit and J GThompson ldquoSolvability of groups of odd orderrdquoPacific Journal of Mathematics vol 13 pp 775ndash1029 1963
[10] R Brauer and K A Fowler ldquoOn groups of even orderrdquo Annalsof Mathematics vol 62 no 3 pp 565ndash583 1955
[11] D Gorenstein The Classification of Finite Ssimple Groups Vol1 Groups of Noncharacteristic 2 Type The University Series inMathematics Plenum Press New York NY USA 1983
[12] D Gorenstein R Lyons and R Solomon The Classification ofthe Finite Simple Groups vol 40 of Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 1994
[13] C Bates and P Rowley ldquoInvolutions in Conwayrsquos largest simplegrouprdquo LMS Journal of Computation andMathematics vol 7 pp337ndash351 2004
[14] P Rowley and P Taylor ldquoPoint-line collinearity graphs of twosporadic minimal parabolic geometriesrdquo Journal of Algebra vol331 pp 304ndash310 2011
[15] P Rowley and P Taylor ldquoInvolutions in Jankorsquos simple group 1198694rdquoLMS Journal of Computation and Mathematics vol 14 pp 238ndash253 2011
[16] P Taylor ldquoInvolutions in Fischerrsquos sporadic groupsrdquo preprinthttpeprintsmamanacuk1622
[17] The GAP Group GAP-Groups Algorithms Programming Ver-sion 43 2002 httpwwwgap-systemorg
[18] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I The user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997
[19] R Wilson P Walsh J Tripp et al ldquoAtlas o f finite grouprepresentationsrdquo httpbrauermathsqmulacukAtlasv3
[20] J H Conway R T Curtis S P Norton R A Parker and R AWilson Atlas of Finite Groups Clarendon Oxford UK 1985
[21] R AWilson ldquoStandard generators for sporadic simple groupsrdquoJournal of Algebra vol 184 no 2 pp 505ndash515 1996
[22] D Gorenstein Finite Groups Chelsea Publishing New YorkNY USA 2nd edition 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Algebra 9
Table 29 Continued
119883119862
|O| 119892
141198627560 119887(119886119887)
7560 119887(119886119887)3
141198637560 119887
3(119886119887)
7560 119887(119886119887)2
15119860 15120 (119886119887)1198872
21119860 15120 (119886119887)4119887(119886119887)
21119861 15120 (119886119887)119887
21119862 15120 (119886119887)3119887
21119863 15120 1198872
281198607560 119887
2(119886119887)119887
2
7560 (119886119887)2
281198617560 119887
4(119886119887)3
7560 (119886119887)119887(119886119887)2
Table 30
119883119862|O| 119892
2119860
640 11988611988731198861198872119886119887119886
480 11988611988721198861198872119886119887119886119887119886119887119886119887119886119887
21198861198872119886
240 11988611988721198861198873119886119887311988611988731198861198873119886119887211988611988711988611988721198861198873119886119887119886119887119886119887119886119887
31198861198873
30 1198861198872119886119887119886119887119886119887211988611988721198861198873119886119887311988611988711988611988711988611988721198861198873119886119887211988611988721198861198873119886119887211988611988711988611988711988611988731198863119860 10240 119886119887
3119886119887119886119887119886119887119886119887119886119887
2119886
4119860 5120 119886119887119886119887119886119887119886
4119862
7680 119886119887119886119887211988611988721198861198872119886119887119886
3840 119886119887211988611988731198861198871198861198872119886119887119886
3840 119886119887119886119887211988611988711988611988731198861198872119886
411986315360 119886119887119886
7680 1198861198872119886119887211988611988711988611988721198861198871198861198872119886
5119860 12288 11988611988731198861198871198861198872119886119887119886119887119886119887119886
5119861 4096 119886119887119886119887119886
611986061440 119886119887
21198861198871198861198872119886119887119886
30720 119886119887119886119887119886119887119886119887119886119887119886
7119860 61440 119886
811986261440 119886119887119886119887
21198861198872119886
61440 11988611988721198861198872119886119887119886
10119860 61440 11988611988721198861198872119886119887119886119887119886119887119886
12119860122880 1198861198872119886119887119886
13119860 61440 1198861198873119886119887119886
312 119870 cong 119867119890
3121119883 = 2119860 119905 = 119886 See Table 27
3122119883 = 2119861 119905 = (1198861198872)6 See Table 28
3123 119883 = 2119862 119905 = 1198893 See Table 29
313 119870 cong 119877119906
3131 119883 = 2119860 119905 = 1198872 See Table 30
3132 119883 = 2119861 119905 = 119886 See Table 31
314 119870 cong 119878119906119911
3141119883 = 2119860 119905 = ((119886119887)21198861198872)6 See Table 32
Table 31
119883119862 |O| 119892
2119860 455 119887119886119887311988611988721198861198873119886119887
2119861
455 11988711988611988731198861198872119886119887119886119887119886119887119886119887119886119887
455 11988711988611988711988611988711988611988711988611988711988611988721198861198873119886119887
1 1198873119886119887311988611988731198861198872119886119887119886119887119886119887
211988611988721198861198873119886119887119886119887211988611988721198861198872
1 1198871198861198872119886119887119886119887311988611988711988611988721198861198872119886119887119886119887211988611988711988611988731198861198873119886119887119886119887119886119887
31198861198871198861198872119886119887119886119887211988611988731198861198871198861198873
3119860 5824 1198872119886119887119886119887119886119887119886119887119886119887
4119860 3640 (119886119887)10119886(119886119887)15119886((119886119887)
5119886)2
4119862 3640 1198873119886119887119886119887119886119887
21198861198872
4119863 3640 119887311988611988711988611988721198861198873119886119887
51198605824 119887
311988611988711988611988721198861198871198861198872
5824 119887211988611988731198861198872119886119887
5119861 29120 1198872119886119887119886119887
6119860 29120 1198872119886119887119886119887119886119887
2119886119887
7119860
29120 11988721198861198872
29120 119887
29120 1198872
1011986029120 119887119886119887119886119887
2119886119887119886119887
29120 119887119886119887119886119887
1011986129120 119887119886119887119886119887
31198861198873119886119887
29120 11988711988611988731198861198873119886119887119886119887
12119860 58240 1198873119886119887119886119887119886119887
2119886119887
13119860
29120 119887119886119887
29120 119887119886119887119886119887119886119887
29120 11988731198861198872119886119887
1411986029120 119887
311988611988721198861198872119886119887119886119887
29120 11988711988611988721198861198872119886119887119886119887119886119887
1411986129120 119887
21198861198872119886119887119886119887119886119887
29120 11988711988611988711988611988711988611988721198861198872
1411986229120 119887119886119887
2119886119887119886119887119886119887119886119887
2
29120 1198872119886119887119886119887119886119887119886119887
2119886119887
15119860 116480 11988731198861198872119886119887119886119887119886119887
2011986058240 119887
31198861198871198861198872
58240 11988721198861198873119886119887
2611986029120 119887119886119887
2
29120 1198872119886119887
2611986129120 119887119886119887
21198861198872
29120 11988721198861198872119886119887
2611986229120 119887
3119886119887119886119887119886119887
29120 11988711988611988711988611988721198861198872
29119860 116480 1198871198861198872119886119887
29119861 116480 1198873119886119887119886119887
Table 32
119883119862
|O| 119892
2119860360 119886
54 ((119886119887)2119886)4119886((119886119887)
2119886)3
3119861 5120 11988721198861198872119886119887119886119887119886119887119886119887119886119887119886
3119862 9216 119887119886119887119886
4119860 1728 119887211988611988721198861198872119886119887119886119887119886119887
4119862 17280 1198872119886119887
5119861 55296 1198872119886119887119886119887119886
6119863 46080 119887
10 Algebra
Table 33119883119862|O| 119892
2119860 315 1198871198861198872119886119887119886119887119886119887
21198861198872119886119887211988611988721198861198872119886119887119886119887119886119887
2119886119887
2119861
630 1198872119886119887119886119887119886119887119886119887
2119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
21198861198872119886119887119886119887211988611988721198861198872119886119887
560 119887119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
2 (1198871198861198872119886)2(1198872119886)2(119887119886(1198872119886)2)2(1198872119886119887119886)2((1198872119886)21198872119886119887119886)21198871198861198872(119886119887)2
3119860 224 (1198871198861198872119886)3119887(119887119886)3(1198872119886)2(119887119886)5119887(119887119886)4(1198872119886)2119887
3119861 2240 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887
3119862 20160 11988711988611988721198861198871198861198872119886119887
41198602520 119887119886119887119886119887119886119887
2119886119887211988611988721198861198872119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886119887119886119887
4119861
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
2520 11988711988611988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
21198861198872119886119887
1260 119887211988611988721198861198871198861198872119886119887119886119887
4119862
5040 11988711988611988721198861198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
2520 11988711988611988711988611988711988611988721198861198872119886119887119886119887119886119887
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
4119863
10080 11988711988611988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
10080 11988711988611988711988611988721198861198872119886119887119886119887211988611988721198861198872119886119887
1120 (119886119887)4((119886119887)4119886)4119886(119886119887)8((119886119887)4119886)2
1120 (119886(1198872119886)4)2(1198872119886)8119886(119886(1198872119886)4)4(1198872119886)4
5119860 20160 119887119886119887119886119887
5119861 40320 1198872119886119887119886119887119886119887
2119886119887
6119860 10080 1198871198861198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
6119863 20160 11988711988611988721198861198872119886119887
611986440320 119887
2119886119887119886119887119886119887
2119886119887119886119887119886119887
40320 1198872119886119887119886119887211988611988721198861198872119886119887119886119887119886119887
7119860 80640 1198872119886119887119886119887119886119887119886119887
811986020160 119887119886119887
2119886119887119886119887211988611988721198861198872119886119887
20160 1198871198861198872119886119887211988611988721198861198871198861198872119886119887
8119861
20160 1198871198861198872119886119887119886119887119886119887119886119887
211988611988721198861198871198861198872119886119887
20160 11988711988611988711988611988721198861198872119886119887
20160 11988711988611988721198861198872119886119887119886119887
20160 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887119886119887
2119886119887
10080 11988721198861198872119886119887
10080 1198872119886119887119886119887
8119862
20160 11988721198861198871198861198872119886119887119886119887119886119887
20160 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
2119886119887
20160 1198872119886119887211988611988721198861198871198861198872119886119887
20160 1198871198861198872119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
10119860
40320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
119887211988611988711988611988721198861198871198861198872119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886119887
20160 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887
20160 11988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
1011986140320 119887119886119887119886119887119886119887119886119887
2119886119887
40320 1198871198861198872119886119887119886119887119886119887119886119887
11119860 161280 11988721198861198871198861198872119886119887119886119887
1211986040320 119887
2119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
20160 119887211988611988711988611988721198861198872119886119887119886119887119886119887
2119886119887
1211986140320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886119887
11988721198861198872119886119887211988611988721198861198872119886119887
12119862 40320 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
1211986380640 119887119886119887
2119886119887119886119887119886119887
80640 1198871198861198871198861198871198861198872119886119887
13119860 161280 119887119886119887
13119861 161280 11988721198861198872119886119887119886119887
14119860 80640 119887211988611988711988611988721198861198872119886119887119886119887
15119860 161280 119887
15119861 161280 1198872119886119887
15119862 161280 119887211988611988721198861198871198861198872119886119887119886119887119886119887
Table 33 Continued119883119862
|O| 119892
2011986080640
80640
1003816100381610038161003816100381610038161003816100381610038161003816
119887119886119887211988611988721198861198872119886119887119886119887
119887119886119887119886119887211988611988721198861198872119886119887
21119860 161280 119887119886119887211988611988721198861198872119886119887119886119887119886119887
21119861 161280 119887119886119887211988611988721198861198872119886119887
2411986080640 119887119886119887119886119887
2119886119887
80640 1198871198861198872119886119887119886119887
Table 34
119883119862
|O| 119892
2119860 315 ((119886119887)3119886)2119886((119886119887)
3119886)4119886(119886119887)
3((119886119887)3119886)2119886
2119861 1800 1198861198871198861198871198861198872
3119860 200 119887211988611988721198861198871198861198872119886119887119886119887
3119862 16800 11988611988721198861198872119886119887119886119887119886
41198603150 119887119886119887
21198861198871198861198872119886
630 1198871198861198872119886119887119886119887119886119887119886119887119886119887
2119886
4119862 12600 1198871198861198872119886119887119886
5119861 20160 119886119887119886
6119860 12600 1198861198871198861198871198861198872119886119887
6119864 100800 119887119886
7119860 100800 119887
811986025200 119886119887119886119887119886119887
25200 119887119886119887119886119887119886119887
12119860 50400 119886
Table 35
119883119862
|O| 119892
2119860 495 (119886119887)5119886 (119886119887)
2(119886 (119886119887)
5)4
(119886119887)5
2119861 792 11988721198861198871198861198872119886119887119886119887119886119887
2119886119887119886119887
3119860 264 119886(119886119887)9((119886119887)9119886)4119886(119886119887)
9((119886119887)9119886)2
31198615280 119886(119886119887)
6((119886119887)3119886)4
1760 (119886((119886119887)3119886)2)4119886
3119862 2640 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886
4119860
2970 11988711988611988711988611988721198861198871198861198872119886119887119886
495
495
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886
4119861 11880 119887119886119887211988611988721198861198871198861198872119886119887119886
4119862 11880 1198872119886119887
5119860 19008 119887119886119887119886
5119861 47520 1198861198872119886119887119886119887119886
6119860
11880 119886119887119886
3960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198871198861198872119886
11988611988711988611988721198861198872119886
611986115840 119886119887
2119886119887119886119887
15840 1198861198872119886119887119886119887119886119887119886119887119886119887
611986215840 119887
21198861198872119886119887119886
15840 1198872119886119887119886119887119886119887119886119887119886119887
21198861198872
611986347520 119886119887
2119886119887119886119887119886119887119886
15840 1198861198872119886119887119886119887119886119887
2119886119887119886
Algebra 11
Table 35 Continued
119883119862
|O| 119892
6119864 15840 119886119887119886119887119886119887119886119887211988611988711988611988721198861198872119886119887
7119860 47520 1198871198861198872119886119887119886
811986147520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886
119886119887119886119887119886119887
911986095040 119887119886119887119886119887119886119887
31680 119887119886119887119886119887211988611988721198861198871198861198872119886
911986195040 119887
2119886119887119886119887119886
31680 1198861198872119886119887119886119887119886119887119886119887119886119887119886
10119860 95040 119886119887
1011986147520 119887119886119887119886119887
21198861198872119886
47520 11988711988611988721198861198872119886119887119886
11119860 190080 119886
12119860 47520 1198872119886119887119886
12119861
47520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198872119886119887119886119887119886119887
2
119886119887119886119887119886119887119886119887119886
15840
15840
1003816100381610038161003816100381610038161003816100381610038161003816
11988721198861198872119886119887119886119887
119887119886119887119886119887119886
1211986247520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198872119886119887119886
1198861198871198861198871198861198871198861198871198861198871198861198872
12119864
47520 11988721198861198872119886
47520 119886119887119886119887
47520 119887119886119887119886119887119886119887119886119887
47520 1198872119886119887119886119887119886119887119886
1411986047520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198871198861198872119886119887119886119887
119887
15119862 190080 119886119887119886119887119886
1811986095040 119887119886119887
2119886119887119886119887119886
95040 11988611988721198861198872119886119887119886119887
1811986195040 119887
21198861198872119886119887119886119887119886
95040 1198871198861198871198861198871198861198872119886
2011986095040
95040
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198872119886119887119886
119887119886
Table 36
119883119862
|O| 119892
2119860 630 119886119887211988621198872119886
3119861 8960 11988611988721198862
3119862 1920 11988621198871198862
Table 37
119883119862
|O| 119892
4119861 30240 1198861198872119886
5119861 48384 1198861198871198862
6119862 80640 119886
Table 38
119883119862
|O| 119892
2119860 495 119887119886119887119886211988711988621198871198861198872119886119887
2119861396 119886119887
311988611988721198862119887119886
396 11988621198873119886119887211988621198871198862
31198601320 119887119886
211988711988621198872119886119887
440 11988611988721198861198871198861198872119886119887119886119887
31198612640 119886
211988721198861198871198861198873119886
1584 119886119887119886119887119886211988731198862
3119862 7920 11988711988611988721198861198872119886119887
4119860 2970 1198871198861198871198861198871198861198872119886119887119886
4119861
11880 1198862119887
5940 1198872
2970 1198871198861198871198861198872119886119887
51198609504 119887
21198861198871198862
9504 119887119886211988731198862119887119886
5119861 47520 11988621198872119886
611986011880 119887119886
211988721198861198872119886
3960 119886119887211988621198871198861198871198862119887119886
611986115840 119886
2
15840 119886
6119862
23760 1198871198861198871198861198873119886
23760 11988711988621198872119886119887119886
15840 1198871198861198873119886119887119886
611986423760 119887
2119886
23760 11988621198872
711986095040 119887119886
2
31680 1198871198861198872119886119887119886
811986023760 119887119886119887119886119887
2119886
23760 1198872119886119887119886119887119886119887
811986123760 119886119887119886119887
2119886
23760 11988721198862119887119886
3142 119883 = 2119861 119905 = 119886 See Table 33
3143 119883 = 2119862 119905 = 119888 See Table 34
3144 119883 = 2119863 119905 = (119888(119888119889)12)11 See Table 35
315 119870 cong 1198621199003
3151 119883 = 2119860 119905 = 1198872 See Tables 36 and 37
12 Algebra
Table 39
119883119862
|O| 119892
8119862
23760 11988611988721198861198872
23760 1198872119886211988721198862
23760 1198872119886119887119886
23760 11988711988621198871198861198873
911986031680 119887119886119887
21198861198873
31680 119886119887119886119887119886119887119886119887
9119861 63360 1198861198871198861198871198861198872
1011986047520 119886119887119886
21198872
47520 1198871198861198871198861198872
1011986147520 119886119887119886
47520 119886211988731198862
1211986023760 119886
21198871198862
23760 1198861198873119886
12119861
47520 119887
47520 1198861198871198861198871198862
47520 1198871198861198872119886119887
1411986095040 119887119886119887119886
95040 1198861198872119886119887119886
1511986095040 119886119887
2
95040 119886119887119886119887
15119861 190080 1198861198871198861198872
1811986095040 119886119887
2119886119887
95040 119887119886119887119886119887
21119860 190080 11988711988621198871198862
2411986095040 119886119887
2119886
95040 119886211988721198862
2411986195040 119886
2119887119886
95040 119887119886
3011986095040 119886119887119886
2
95040 11988611988731198862
3152 119883 = 2119861 119905 = ((1198862119887)3119886119887)5 See Tables 38 and 39
316 119870 cong 1198621199002
3161 119883 = 2119860 119905 = 119886 See Table 40
3162 119883 = 2119861 119905 = 119886(119886119887119886119887119886)2119886(119886119887119886119887119886)7 See Table 41
3163 119883 = 2119862 119905 = (119887minus1(1198861198872)2119886119887minus1(1198861198872)4)5 See Table 42
Table 40
119883119862
|O| 119892
2119861 1260 1198871198861198872119886119887
2119862 1008 11988731198861198871198861198873
3119861 14336 119887119886119887
4119862 40320 119887
Table 41
119883119862
|O| 119892
2119860 15 (1198874119886)2119887311988611988721198861198874(119887119886119887)
2119886(119887119886119887)
711988611988711988611988731198861198872(119886119887)2
21198611680 119887
31198861198871198861198873
210 11988741198861198874119886119887119886119887
2119862 2520 (119886119887)5119886(119886119887)
2119886(119886119887)
3
3119861 35840 1198872119886119887
4119860 1920 11988711988611988731198861198872119886119887119886119887
4119862 20160 1198871198861198872119886119887
4119864 161280 119887119886119887
411986513440 119887119886119887119886119887119886119887119886119887
13440 119887311988611988711988611988721198861198872
5119861 344064 1198872
6119864 430080 119887
Table 42
119883119862
|O| 119892
2119860240 119886119887
4119886119887211988611988711988611988741198861198873119886119887311988611988741198861198873119886119887119886
45 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887411988611988741198861198872119886
2119861
720 1198872119886
240 11988741198861198874119886119887311988611988741198861198874119886119887211988611988731198861198872119886
180 119886119887211988611988731198861198872119886119887119886119887311988611988731198861198872
30 1198861198871198861198874119886
2119862
1440 1198861198871198861198873119886119887311988611988721198861198871198861198872119886
1440 119886119887311988611988741198861198873119886119887211988611988721198861198874119886
1440 1198872119886119887119886119887119886119887
311988611988741198861198873
1152 11988611988711988611988731198861198873119886119887311988611988741198861198872119886
360 11988611988711988611988711988611988731198861198872119886119887119886119887311988611988731198861198872
3119860 10240 1198861198874119886119887211988611988711988611988721198861198871198861198872119886
3119861
15360 11988611988731198861198872119886
7680 11988731198861198872
1024 1198861198872119886119887211988611988731198861198873119886
41198603840 119887
411988611988741198861198873119886119887411988611988741198861198874
1440 1198861198871198861198872119886119887211988611988741198861198872
4119861
11520 1198871198861198874119886119887411988611988731198861198873119886
5760 119886119887211988611988711988611988721198861198871198861198873119886
5760 11988611988721198861198874119886119887311988611988741198861198873119886
1440 11988611988741198861198871198861198872119886119887411988611988721198861198871198861198873119886
1440 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887119886
720 119886119887119886119887411988611988721198861198872119886119887211988611988731198861198873
720 1198872119886119887411988611988721198861198872119886119887411988611988731198861198874119886
Algebra 13
Table 42 Continued
119883119862
|O| 119892
4119862
11520 11988611988721198861198873119886
11520 1198861198873119886119887119886119887311988611988721198861198872
5760 119887311988611988741198861198872119886
5760 11988611988731198861198871198861198872
2880 11988611988711988611988741198861198872119886
1440 119887411988611988741198861198873119886119887211988611988721198861198873119886
1440 1198861198872119886119887311988611988731198861198872119886119887119886119887
720 11988611988711988611988711988611988711988611988721198861198871198861198873119886
720 119886119887311988611988741198861198872119886
4119863
23040 1198871198861198873119886119887119886119887211988611988731198861198872
11520 119886119887119886119887311988611988731198861198871198861198873
5760 119886119887411988611988711988611988741198861198872119886
2880 11988611988711988611988711988611988741198861198872119886119887411988611988741198861198873119886
2880 1198861198872119886119887119886119887119886119887
311988611988711988611988741198861198874119886
4119864
23040 1198871198861198871198861198871198861198872
23040 119886119887311988611988721198861198872
11520 1198861198873119886119887211988611988741198861198872
11520 119887311988611988711988611988731198861198872119886
5760 11988731198861198874119886
5760 1198861198871198861198872
5760 1198874119886119887311988611988721198861198871198861198873119886
4119865
23040 119886119887119886119887211988611988721198861198872
23040 1198873119886119887311988611988731198861198874119886
11520 119887411988611988741198861198872119886119887411988611988731198861198872119886
11520 119886119887311988611988741198861198873119886119887211988611988721198861198872
5760 1198872119886119887411988611988721198861198872119886119887211988611988731198861198872119886
5760 1198872119886119887211988611988731198861198873119886119887411988611988721198861198873119886
5760 1198861198873119886119887211988611988741198861198872119886119887311988611988731198861198872119886
5760 1198861198874119886119887119886119887119886119887
3119886119887211988611988721198861198873
5760 119886119887211988611988731198861198871198861198872119886119887211988611988731198861198873
5760 1198874119886119887411988611988721198861198872119886119887211988611988731198861198872119886
2880 119887311988611988721198861198871198861198872
2880 1198873119886119887411988611988731198861198872
1920 11988611988711988611988741198861198872119886119887211988611988741198861198873
1920 1198871198861198873119886119887311988611988711988611988721198861198874
4119866
46080 1198873119886119887311988611988721198861198873119886
46080 1198861198872119886119887311988611988721198861198872
23040 1198861198871198861198872119886119887211988611988731198861198872119886
23040 11988611988731198861198872119886119887311988611988731198861198874119886
5119860 73728 119886119887211988611988731198861198871198861198872119886
5119861184320 119887
21198861198872119886
122880 119887211988611988731198861198872
Table 42 Continued
119883119862
|O| 119892
6119860 92160 1198871198861198873119886119887411988611988731198861198872119886
6119861 122880 119887211988611988741198861198872119886
6119862
46080 1198872119886119887311988611988731198861198872119886
46080 119887311988611988711988611988721198861198872
15360 1198874119886119887211988611988731198861198873119886
15360 119886119887211988611988721198861198873119886119887
6119863
92160 119887411988611988711988611988721198861198874119886
23040 11988741198861198873119886119887411988611988721198861198872
15360 11988611988721198861198872119886119887119886
15360 119886119887411988611988731198861198873119886
6119864
92160 1198871198861198871198861198872
92160 1198871198861198874119886
46080 1198873119886119887211988611988731198861198872119886
46080 11988711988611988721198861198873119886
46080 11988721198861198871198861198874119886
46080 11988611988711988611988741198861198873
15360 119886119887211988611988711988611988741198861198871198861198872119886
6119865
92160 1198861198873119886119887119886
92160 11988611988741198861198872119886
46080 119887311988611988721198861198871198861198872119886
46080 1198861198874119886119887411988611988731198861198873
46080 1198861198873119886119887411988611988731198861198872
46080 11988721198861198872119886119887119886119887119886
7119860 737280 11988741198861198872
811986092160 119887
2119886119887119886
92160 11988611988741198861198873
8119861
92160 119886119887411988611988721198861198872119886119887119886
92160 119886119887411988611988731198861198873119886119887119886
46080 11988611988711988611988731198861198872119886
46080 119886119887311988611988721198861198874119886
23040 119887119886119887311988611988731198861198871198861198873
23040 11988721198861198874119886119887211988611988721198861198874
8119862
92160 1198873119886119887311988611988731198861198874
92160 1198872
92160 1198873
92160 119887119886119887211988611988721198861198872
8119863
92160 119887311988611988731198861198872
92160 11988611988721198861198873119886119887311988611988741198861198872
92160 119887411988611988731198861198873119886
92160 11988711988611988721198861198871198861198873119886
92160 119887311988611988721198861198872
14 Algebra
Table 42 Continued
119883119862
|O| 119892
8119864
92160 1198861198873119886
92160 1198861198872119886
46080 119886119887311988611988711988611988741198861198872
46080 119887311988611988711988611988741198861198872119886
46080 119886119887211988611988721198861198871198861198873
46080 1198872119886119887119886119887119886
46080 1198872119886119887411988611988731198861198873119886
46080 1198872119886119887211988611988731198861198873119886
46080 1198861198872119886119887211988611988731198861198873
46080 119886119887411988611988741198861198873
23040 1198861198874119886119887211988611988721198861198871198861198873
23040 11988741198861198874119886119887211988611988741198861198873
8119865
184320 11988611988741198861198871198861198873119886
184320 11988611988721198861198874119886119887119886
92160 119886119887119886119887311988611988711988611988721198861198872
92160 11988731198861198874119886119887411988611988731198861198872119886
9119860737280 119886119887
41198861198871198861198872
245760 119887211988611988741198861198872
10119860 368640 119887211988611988731198861198872119886
10119861368640 119886119887
21198861198872
368640 11988731198861198873119886
10119862368640 119886119887
4
368640 11988611988711988611988731198861198871198861198873119886
11119860 1474560 11988721198861198873119886
12119860122880 119887119886119887
21198861198871198861198874119886
122880 119887411988611988711988611988741198861198873
12119861368640 119887
211988611988711988611988731198861198872
184320 1198861198874119886119887211988611988741198861198872
12119862368640 119886119887
2119886119887211988611988731198861198872
184320 119886119887411988611988711988611988721198861198872
12119863
184320 119887411988611988711988611988721198861198872
184320 11988611988711988611988711988611988731198861198873119886
92160 119886119887411988611988721198861198873119886
92160 11988611988721198861198873119886119887119886
12119864368640 119887
311988611988721198861198874
368640 11988711988611988731198861198872
12119865
184320 11988721198861198871198861198872119886
184320 11988611988731198861198874119886
184320 11988611988711988611988731198861198873
184320 1198861198871198861198872119886
92160 1198861198872119886119887119886
92160 11988611988741198861198873119886
12119866 368640 11988711988611988741198861198872119886
Table 42 Continued
119883119862
|O| 119892
12119867
184320 1198874119886119887311988611988731198861198872
184320 119886119887311988611988721198861198873119886
184320 11988611988711988611988711988611988741198861198872119886
184320 1198861198874119886119887311988611988741198861198872119886
184320 119887311988611988721198861198872119886119887
184320 119886119887211988611988731198861198872119886
14119860 737280 1198861198874119886
15119860737280 119887
4119886
737280 1198871198861198872119886
16119860
368640 11988611988731198861198872
368640 11988611988721198861198871198861198874
368640 11988731198861198872119886
368640 11988711988611988741198861198873119886
18119860 737280 11988731198861198873
20119860737280 119886119887
41198861198872
737280 1198873119886119887119886
20119861737280 119886119887
41198861198874119886
737280 119886119887119886119887119886
24119860737280 119886119887
2
737280 1198873119886
24119861737280 119887
4
737280 119887
28119860737280 119886119887119886119887
31198861198872
737280 119886119887411988611988721198861198872
30119860737280 119887
21198861198874
737280 1198871198861198873
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R H Bing ldquoA homeomorphism between the 3-sphere andthe sum of two solid horned spheresrdquo Annals of MathematicsSecond Series vol 56 pp 354ndash362 1952
[2] D Montgomery and L Zippin ldquoExamples of transformationgroupsrdquo Proceedings of the American Mathematical Society vol5 pp 460ndash465 1954
[3] J W Morgan and H Bass Eds The Smith Conjecture vol 112of Pure and Applied Mathematics Academic Press OrlandoFla USA 1984 Papers presented at the symposium held atColumbia University New York NY USA 1979
[4] C M Edwards and G T Ruttimann ldquoInvolutive and Peircegradings in JBWlowast-triplesrdquo Communications in Algebra vol 31no 6 pp 2819ndash2848 2003
Algebra 15
[5] J Gornicki and B E Rhoades ldquoA general fixed point theoremfor involutionsrdquo Indian Journal of Pure and Applied Mathemat-ics vol 27 no 1 pp 13ndash23 1996
[6] S Montgomery Fixed Rings of Finite Automorphism Groupsof Associative Rings vol 818 of Lecture Notes in MathematicsSpringer Berlin Germany 1980
[7] T A Springer ldquoThe classification of involutions of simplealgebraic groupsrdquo Journal of the Faculty of Science University ofTokyo Section IA Mathematics vol 34 no 3 pp 655ndash670 1987
[8] T A Springer ldquoSome results on algebraic group s with invo-lutionsrdquo in Algebraic Groups and Related Topics (KyotoNagoya1983) vol 6 of Advanced Studies in Pure Mathematics pp 525ndash543 North-Holland Amsterdam The Netherlands 1985
[9] W Feit and J GThompson ldquoSolvability of groups of odd orderrdquoPacific Journal of Mathematics vol 13 pp 775ndash1029 1963
[10] R Brauer and K A Fowler ldquoOn groups of even orderrdquo Annalsof Mathematics vol 62 no 3 pp 565ndash583 1955
[11] D Gorenstein The Classification of Finite Ssimple Groups Vol1 Groups of Noncharacteristic 2 Type The University Series inMathematics Plenum Press New York NY USA 1983
[12] D Gorenstein R Lyons and R Solomon The Classification ofthe Finite Simple Groups vol 40 of Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 1994
[13] C Bates and P Rowley ldquoInvolutions in Conwayrsquos largest simplegrouprdquo LMS Journal of Computation andMathematics vol 7 pp337ndash351 2004
[14] P Rowley and P Taylor ldquoPoint-line collinearity graphs of twosporadic minimal parabolic geometriesrdquo Journal of Algebra vol331 pp 304ndash310 2011
[15] P Rowley and P Taylor ldquoInvolutions in Jankorsquos simple group 1198694rdquoLMS Journal of Computation and Mathematics vol 14 pp 238ndash253 2011
[16] P Taylor ldquoInvolutions in Fischerrsquos sporadic groupsrdquo preprinthttpeprintsmamanacuk1622
[17] The GAP Group GAP-Groups Algorithms Programming Ver-sion 43 2002 httpwwwgap-systemorg
[18] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I The user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997
[19] R Wilson P Walsh J Tripp et al ldquoAtlas o f finite grouprepresentationsrdquo httpbrauermathsqmulacukAtlasv3
[20] J H Conway R T Curtis S P Norton R A Parker and R AWilson Atlas of Finite Groups Clarendon Oxford UK 1985
[21] R AWilson ldquoStandard generators for sporadic simple groupsrdquoJournal of Algebra vol 184 no 2 pp 505ndash515 1996
[22] D Gorenstein Finite Groups Chelsea Publishing New YorkNY USA 2nd edition 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Algebra
Table 33119883119862|O| 119892
2119860 315 1198871198861198872119886119887119886119887119886119887
21198861198872119886119887211988611988721198861198872119886119887119886119887119886119887
2119886119887
2119861
630 1198872119886119887119886119887119886119887119886119887
2119886119887119886119887119886119887
21198861198872119886119887119886119887119886119887
21198861198872119886119887119886119887211988611988721198861198872119886119887
560 119887119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
2 (1198871198861198872119886)2(1198872119886)2(119887119886(1198872119886)2)2(1198872119886119887119886)2((1198872119886)21198872119886119887119886)21198871198861198872(119886119887)2
3119860 224 (1198871198861198872119886)3119887(119887119886)3(1198872119886)2(119887119886)5119887(119887119886)4(1198872119886)2119887
3119861 2240 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887
3119862 20160 11988711988611988721198861198871198861198872119886119887
41198602520 119887119886119887119886119887119886119887
2119886119887211988611988721198861198872119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886119887119886119887
4119861
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
2520 11988711988611988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
1260 1198872119886119887119886119887119886119887
21198861198872119886119887
1260 119887211988611988721198861198871198861198872119886119887119886119887
4119862
5040 11988711988611988721198861198871198861198872119886119887119886119887119886119887
21198861198871198861198872119886119887
2520 11988711988611988711988611988711988611988721198861198872119886119887119886119887119886119887
2520 11988721198861198872119886119887119886119887119886119887
211988611988711988611988721198861198872119886119887119886119887
4119863
10080 11988711988611988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
10080 11988711988611988711988611988721198861198872119886119887119886119887211988611988721198861198872119886119887
1120 (119886119887)4((119886119887)4119886)4119886(119886119887)8((119886119887)4119886)2
1120 (119886(1198872119886)4)2(1198872119886)8119886(119886(1198872119886)4)4(1198872119886)4
5119860 20160 119887119886119887119886119887
5119861 40320 1198872119886119887119886119887119886119887
2119886119887
6119860 10080 1198871198861198871198861198871198861198871198861198872119886119887119886119887119886119887119886119887
6119863 20160 11988711988611988721198861198872119886119887
611986440320 119887
2119886119887119886119887119886119887
2119886119887119886119887119886119887
40320 1198872119886119887119886119887211988611988721198861198872119886119887119886119887119886119887
7119860 80640 1198872119886119887119886119887119886119887119886119887
811986020160 119887119886119887
2119886119887119886119887211988611988721198861198872119886119887
20160 1198871198861198872119886119887211988611988721198861198871198861198872119886119887
8119861
20160 1198871198861198872119886119887119886119887119886119887119886119887
211988611988721198861198871198861198872119886119887
20160 11988711988611988711988611988721198861198872119886119887
20160 11988711988611988721198861198872119886119887119886119887
20160 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887119886119887
2119886119887
10080 11988721198861198872119886119887
10080 1198872119886119887119886119887
8119862
20160 11988721198861198871198861198872119886119887119886119887119886119887
20160 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887119886119887
2119886119887
20160 1198872119886119887211988611988721198861198871198861198872119886119887
20160 1198871198861198872119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
10119860
40320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
119887211988611988711988611988721198861198871198861198872119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886119887
20160 11988721198861198872119886119887119886119887119886119887
2119886119887119886119887119886119887
20160 11988721198861198872119886119887211988611988711988611988721198861198872119886119887119886119887
1011986140320 119887119886119887119886119887119886119887119886119887
2119886119887
40320 1198871198861198872119886119887119886119887119886119887119886119887
11119860 161280 11988721198861198871198861198872119886119887119886119887
1211986040320 119887
2119886119887211988611988711988611988721198861198871198861198872119886119887119886119887
20160 119887211988611988711988611988721198861198872119886119887119886119887119886119887
2119886119887
1211986140320
40320
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886119887
11988721198861198872119886119887211988611988721198861198872119886119887
12119862 40320 119887119886119887211988611988711988611988721198861198872119886119887119886119887119886119887
1211986380640 119887119886119887
2119886119887119886119887119886119887
80640 1198871198861198871198861198871198861198872119886119887
13119860 161280 119887119886119887
13119861 161280 11988721198861198872119886119887119886119887
14119860 80640 119887211988611988711988611988721198861198872119886119887119886119887
15119860 161280 119887
15119861 161280 1198872119886119887
15119862 161280 119887211988611988721198861198871198861198872119886119887119886119887119886119887
Table 33 Continued119883119862
|O| 119892
2011986080640
80640
1003816100381610038161003816100381610038161003816100381610038161003816
119887119886119887211988611988721198861198872119886119887119886119887
119887119886119887119886119887211988611988721198861198872119886119887
21119860 161280 119887119886119887211988611988721198861198872119886119887119886119887119886119887
21119861 161280 119887119886119887211988611988721198861198872119886119887
2411986080640 119887119886119887119886119887
2119886119887
80640 1198871198861198872119886119887119886119887
Table 34
119883119862
|O| 119892
2119860 315 ((119886119887)3119886)2119886((119886119887)
3119886)4119886(119886119887)
3((119886119887)3119886)2119886
2119861 1800 1198861198871198861198871198861198872
3119860 200 119887211988611988721198861198871198861198872119886119887119886119887
3119862 16800 11988611988721198861198872119886119887119886119887119886
41198603150 119887119886119887
21198861198871198861198872119886
630 1198871198861198872119886119887119886119887119886119887119886119887119886119887
2119886
4119862 12600 1198871198861198872119886119887119886
5119861 20160 119886119887119886
6119860 12600 1198861198871198861198871198861198872119886119887
6119864 100800 119887119886
7119860 100800 119887
811986025200 119886119887119886119887119886119887
25200 119887119886119887119886119887119886119887
12119860 50400 119886
Table 35
119883119862
|O| 119892
2119860 495 (119886119887)5119886 (119886119887)
2(119886 (119886119887)
5)4
(119886119887)5
2119861 792 11988721198861198871198861198872119886119887119886119887119886119887
2119886119887119886119887
3119860 264 119886(119886119887)9((119886119887)9119886)4119886(119886119887)
9((119886119887)9119886)2
31198615280 119886(119886119887)
6((119886119887)3119886)4
1760 (119886((119886119887)3119886)2)4119886
3119862 2640 1198872119886119887119886119887119886119887
2119886119887119886119887119886119887
2119886
4119860
2970 11988711988611988711988611988721198861198871198861198872119886119887119886
495
495
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887
1198871198861198871198861198871198861198872119886119887119886119887119886
4119861 11880 119887119886119887211988611988721198861198871198861198872119886119887119886
4119862 11880 1198872119886119887
5119860 19008 119887119886119887119886
5119861 47520 1198861198872119886119887119886119887119886
6119860
11880 119886119887119886
3960
3960
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198871198861198872119886
11988611988711988611988721198861198872119886
611986115840 119886119887
2119886119887119886119887
15840 1198861198872119886119887119886119887119886119887119886119887119886119887
611986215840 119887
21198861198872119886119887119886
15840 1198872119886119887119886119887119886119887119886119887119886119887
21198861198872
611986347520 119886119887
2119886119887119886119887119886119887119886
15840 1198861198872119886119887119886119887119886119887
2119886119887119886
Algebra 11
Table 35 Continued
119883119862
|O| 119892
6119864 15840 119886119887119886119887119886119887119886119887211988611988711988611988721198861198872119886119887
7119860 47520 1198871198861198872119886119887119886
811986147520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886
119886119887119886119887119886119887
911986095040 119887119886119887119886119887119886119887
31680 119887119886119887119886119887211988611988721198861198871198861198872119886
911986195040 119887
2119886119887119886119887119886
31680 1198861198872119886119887119886119887119886119887119886119887119886119887119886
10119860 95040 119886119887
1011986147520 119887119886119887119886119887
21198861198872119886
47520 11988711988611988721198861198872119886119887119886
11119860 190080 119886
12119860 47520 1198872119886119887119886
12119861
47520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198872119886119887119886119887119886119887
2
119886119887119886119887119886119887119886119887119886
15840
15840
1003816100381610038161003816100381610038161003816100381610038161003816
11988721198861198872119886119887119886119887
119887119886119887119886119887119886
1211986247520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198872119886119887119886
1198861198871198861198871198861198871198861198871198861198871198861198872
12119864
47520 11988721198861198872119886
47520 119886119887119886119887
47520 119887119886119887119886119887119886119887119886119887
47520 1198872119886119887119886119887119886119887119886
1411986047520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198871198861198872119886119887119886119887
119887
15119862 190080 119886119887119886119887119886
1811986095040 119887119886119887
2119886119887119886119887119886
95040 11988611988721198861198872119886119887119886119887
1811986195040 119887
21198861198872119886119887119886119887119886
95040 1198871198861198871198861198871198861198872119886
2011986095040
95040
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198872119886119887119886
119887119886
Table 36
119883119862
|O| 119892
2119860 630 119886119887211988621198872119886
3119861 8960 11988611988721198862
3119862 1920 11988621198871198862
Table 37
119883119862
|O| 119892
4119861 30240 1198861198872119886
5119861 48384 1198861198871198862
6119862 80640 119886
Table 38
119883119862
|O| 119892
2119860 495 119887119886119887119886211988711988621198871198861198872119886119887
2119861396 119886119887
311988611988721198862119887119886
396 11988621198873119886119887211988621198871198862
31198601320 119887119886
211988711988621198872119886119887
440 11988611988721198861198871198861198872119886119887119886119887
31198612640 119886
211988721198861198871198861198873119886
1584 119886119887119886119887119886211988731198862
3119862 7920 11988711988611988721198861198872119886119887
4119860 2970 1198871198861198871198861198871198861198872119886119887119886
4119861
11880 1198862119887
5940 1198872
2970 1198871198861198871198861198872119886119887
51198609504 119887
21198861198871198862
9504 119887119886211988731198862119887119886
5119861 47520 11988621198872119886
611986011880 119887119886
211988721198861198872119886
3960 119886119887211988621198871198861198871198862119887119886
611986115840 119886
2
15840 119886
6119862
23760 1198871198861198871198861198873119886
23760 11988711988621198872119886119887119886
15840 1198871198861198873119886119887119886
611986423760 119887
2119886
23760 11988621198872
711986095040 119887119886
2
31680 1198871198861198872119886119887119886
811986023760 119887119886119887119886119887
2119886
23760 1198872119886119887119886119887119886119887
811986123760 119886119887119886119887
2119886
23760 11988721198862119887119886
3142 119883 = 2119861 119905 = 119886 See Table 33
3143 119883 = 2119862 119905 = 119888 See Table 34
3144 119883 = 2119863 119905 = (119888(119888119889)12)11 See Table 35
315 119870 cong 1198621199003
3151 119883 = 2119860 119905 = 1198872 See Tables 36 and 37
12 Algebra
Table 39
119883119862
|O| 119892
8119862
23760 11988611988721198861198872
23760 1198872119886211988721198862
23760 1198872119886119887119886
23760 11988711988621198871198861198873
911986031680 119887119886119887
21198861198873
31680 119886119887119886119887119886119887119886119887
9119861 63360 1198861198871198861198871198861198872
1011986047520 119886119887119886
21198872
47520 1198871198861198871198861198872
1011986147520 119886119887119886
47520 119886211988731198862
1211986023760 119886
21198871198862
23760 1198861198873119886
12119861
47520 119887
47520 1198861198871198861198871198862
47520 1198871198861198872119886119887
1411986095040 119887119886119887119886
95040 1198861198872119886119887119886
1511986095040 119886119887
2
95040 119886119887119886119887
15119861 190080 1198861198871198861198872
1811986095040 119886119887
2119886119887
95040 119887119886119887119886119887
21119860 190080 11988711988621198871198862
2411986095040 119886119887
2119886
95040 119886211988721198862
2411986195040 119886
2119887119886
95040 119887119886
3011986095040 119886119887119886
2
95040 11988611988731198862
3152 119883 = 2119861 119905 = ((1198862119887)3119886119887)5 See Tables 38 and 39
316 119870 cong 1198621199002
3161 119883 = 2119860 119905 = 119886 See Table 40
3162 119883 = 2119861 119905 = 119886(119886119887119886119887119886)2119886(119886119887119886119887119886)7 See Table 41
3163 119883 = 2119862 119905 = (119887minus1(1198861198872)2119886119887minus1(1198861198872)4)5 See Table 42
Table 40
119883119862
|O| 119892
2119861 1260 1198871198861198872119886119887
2119862 1008 11988731198861198871198861198873
3119861 14336 119887119886119887
4119862 40320 119887
Table 41
119883119862
|O| 119892
2119860 15 (1198874119886)2119887311988611988721198861198874(119887119886119887)
2119886(119887119886119887)
711988611988711988611988731198861198872(119886119887)2
21198611680 119887
31198861198871198861198873
210 11988741198861198874119886119887119886119887
2119862 2520 (119886119887)5119886(119886119887)
2119886(119886119887)
3
3119861 35840 1198872119886119887
4119860 1920 11988711988611988731198861198872119886119887119886119887
4119862 20160 1198871198861198872119886119887
4119864 161280 119887119886119887
411986513440 119887119886119887119886119887119886119887119886119887
13440 119887311988611988711988611988721198861198872
5119861 344064 1198872
6119864 430080 119887
Table 42
119883119862
|O| 119892
2119860240 119886119887
4119886119887211988611988711988611988741198861198873119886119887311988611988741198861198873119886119887119886
45 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887411988611988741198861198872119886
2119861
720 1198872119886
240 11988741198861198874119886119887311988611988741198861198874119886119887211988611988731198861198872119886
180 119886119887211988611988731198861198872119886119887119886119887311988611988731198861198872
30 1198861198871198861198874119886
2119862
1440 1198861198871198861198873119886119887311988611988721198861198871198861198872119886
1440 119886119887311988611988741198861198873119886119887211988611988721198861198874119886
1440 1198872119886119887119886119887119886119887
311988611988741198861198873
1152 11988611988711988611988731198861198873119886119887311988611988741198861198872119886
360 11988611988711988611988711988611988731198861198872119886119887119886119887311988611988731198861198872
3119860 10240 1198861198874119886119887211988611988711988611988721198861198871198861198872119886
3119861
15360 11988611988731198861198872119886
7680 11988731198861198872
1024 1198861198872119886119887211988611988731198861198873119886
41198603840 119887
411988611988741198861198873119886119887411988611988741198861198874
1440 1198861198871198861198872119886119887211988611988741198861198872
4119861
11520 1198871198861198874119886119887411988611988731198861198873119886
5760 119886119887211988611988711988611988721198861198871198861198873119886
5760 11988611988721198861198874119886119887311988611988741198861198873119886
1440 11988611988741198861198871198861198872119886119887411988611988721198861198871198861198873119886
1440 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887119886
720 119886119887119886119887411988611988721198861198872119886119887211988611988731198861198873
720 1198872119886119887411988611988721198861198872119886119887411988611988731198861198874119886
Algebra 13
Table 42 Continued
119883119862
|O| 119892
4119862
11520 11988611988721198861198873119886
11520 1198861198873119886119887119886119887311988611988721198861198872
5760 119887311988611988741198861198872119886
5760 11988611988731198861198871198861198872
2880 11988611988711988611988741198861198872119886
1440 119887411988611988741198861198873119886119887211988611988721198861198873119886
1440 1198861198872119886119887311988611988731198861198872119886119887119886119887
720 11988611988711988611988711988611988711988611988721198861198871198861198873119886
720 119886119887311988611988741198861198872119886
4119863
23040 1198871198861198873119886119887119886119887211988611988731198861198872
11520 119886119887119886119887311988611988731198861198871198861198873
5760 119886119887411988611988711988611988741198861198872119886
2880 11988611988711988611988711988611988741198861198872119886119887411988611988741198861198873119886
2880 1198861198872119886119887119886119887119886119887
311988611988711988611988741198861198874119886
4119864
23040 1198871198861198871198861198871198861198872
23040 119886119887311988611988721198861198872
11520 1198861198873119886119887211988611988741198861198872
11520 119887311988611988711988611988731198861198872119886
5760 11988731198861198874119886
5760 1198861198871198861198872
5760 1198874119886119887311988611988721198861198871198861198873119886
4119865
23040 119886119887119886119887211988611988721198861198872
23040 1198873119886119887311988611988731198861198874119886
11520 119887411988611988741198861198872119886119887411988611988731198861198872119886
11520 119886119887311988611988741198861198873119886119887211988611988721198861198872
5760 1198872119886119887411988611988721198861198872119886119887211988611988731198861198872119886
5760 1198872119886119887211988611988731198861198873119886119887411988611988721198861198873119886
5760 1198861198873119886119887211988611988741198861198872119886119887311988611988731198861198872119886
5760 1198861198874119886119887119886119887119886119887
3119886119887211988611988721198861198873
5760 119886119887211988611988731198861198871198861198872119886119887211988611988731198861198873
5760 1198874119886119887411988611988721198861198872119886119887211988611988731198861198872119886
2880 119887311988611988721198861198871198861198872
2880 1198873119886119887411988611988731198861198872
1920 11988611988711988611988741198861198872119886119887211988611988741198861198873
1920 1198871198861198873119886119887311988611988711988611988721198861198874
4119866
46080 1198873119886119887311988611988721198861198873119886
46080 1198861198872119886119887311988611988721198861198872
23040 1198861198871198861198872119886119887211988611988731198861198872119886
23040 11988611988731198861198872119886119887311988611988731198861198874119886
5119860 73728 119886119887211988611988731198861198871198861198872119886
5119861184320 119887
21198861198872119886
122880 119887211988611988731198861198872
Table 42 Continued
119883119862
|O| 119892
6119860 92160 1198871198861198873119886119887411988611988731198861198872119886
6119861 122880 119887211988611988741198861198872119886
6119862
46080 1198872119886119887311988611988731198861198872119886
46080 119887311988611988711988611988721198861198872
15360 1198874119886119887211988611988731198861198873119886
15360 119886119887211988611988721198861198873119886119887
6119863
92160 119887411988611988711988611988721198861198874119886
23040 11988741198861198873119886119887411988611988721198861198872
15360 11988611988721198861198872119886119887119886
15360 119886119887411988611988731198861198873119886
6119864
92160 1198871198861198871198861198872
92160 1198871198861198874119886
46080 1198873119886119887211988611988731198861198872119886
46080 11988711988611988721198861198873119886
46080 11988721198861198871198861198874119886
46080 11988611988711988611988741198861198873
15360 119886119887211988611988711988611988741198861198871198861198872119886
6119865
92160 1198861198873119886119887119886
92160 11988611988741198861198872119886
46080 119887311988611988721198861198871198861198872119886
46080 1198861198874119886119887411988611988731198861198873
46080 1198861198873119886119887411988611988731198861198872
46080 11988721198861198872119886119887119886119887119886
7119860 737280 11988741198861198872
811986092160 119887
2119886119887119886
92160 11988611988741198861198873
8119861
92160 119886119887411988611988721198861198872119886119887119886
92160 119886119887411988611988731198861198873119886119887119886
46080 11988611988711988611988731198861198872119886
46080 119886119887311988611988721198861198874119886
23040 119887119886119887311988611988731198861198871198861198873
23040 11988721198861198874119886119887211988611988721198861198874
8119862
92160 1198873119886119887311988611988731198861198874
92160 1198872
92160 1198873
92160 119887119886119887211988611988721198861198872
8119863
92160 119887311988611988731198861198872
92160 11988611988721198861198873119886119887311988611988741198861198872
92160 119887411988611988731198861198873119886
92160 11988711988611988721198861198871198861198873119886
92160 119887311988611988721198861198872
14 Algebra
Table 42 Continued
119883119862
|O| 119892
8119864
92160 1198861198873119886
92160 1198861198872119886
46080 119886119887311988611988711988611988741198861198872
46080 119887311988611988711988611988741198861198872119886
46080 119886119887211988611988721198861198871198861198873
46080 1198872119886119887119886119887119886
46080 1198872119886119887411988611988731198861198873119886
46080 1198872119886119887211988611988731198861198873119886
46080 1198861198872119886119887211988611988731198861198873
46080 119886119887411988611988741198861198873
23040 1198861198874119886119887211988611988721198861198871198861198873
23040 11988741198861198874119886119887211988611988741198861198873
8119865
184320 11988611988741198861198871198861198873119886
184320 11988611988721198861198874119886119887119886
92160 119886119887119886119887311988611988711988611988721198861198872
92160 11988731198861198874119886119887411988611988731198861198872119886
9119860737280 119886119887
41198861198871198861198872
245760 119887211988611988741198861198872
10119860 368640 119887211988611988731198861198872119886
10119861368640 119886119887
21198861198872
368640 11988731198861198873119886
10119862368640 119886119887
4
368640 11988611988711988611988731198861198871198861198873119886
11119860 1474560 11988721198861198873119886
12119860122880 119887119886119887
21198861198871198861198874119886
122880 119887411988611988711988611988741198861198873
12119861368640 119887
211988611988711988611988731198861198872
184320 1198861198874119886119887211988611988741198861198872
12119862368640 119886119887
2119886119887211988611988731198861198872
184320 119886119887411988611988711988611988721198861198872
12119863
184320 119887411988611988711988611988721198861198872
184320 11988611988711988611988711988611988731198861198873119886
92160 119886119887411988611988721198861198873119886
92160 11988611988721198861198873119886119887119886
12119864368640 119887
311988611988721198861198874
368640 11988711988611988731198861198872
12119865
184320 11988721198861198871198861198872119886
184320 11988611988731198861198874119886
184320 11988611988711988611988731198861198873
184320 1198861198871198861198872119886
92160 1198861198872119886119887119886
92160 11988611988741198861198873119886
12119866 368640 11988711988611988741198861198872119886
Table 42 Continued
119883119862
|O| 119892
12119867
184320 1198874119886119887311988611988731198861198872
184320 119886119887311988611988721198861198873119886
184320 11988611988711988611988711988611988741198861198872119886
184320 1198861198874119886119887311988611988741198861198872119886
184320 119887311988611988721198861198872119886119887
184320 119886119887211988611988731198861198872119886
14119860 737280 1198861198874119886
15119860737280 119887
4119886
737280 1198871198861198872119886
16119860
368640 11988611988731198861198872
368640 11988611988721198861198871198861198874
368640 11988731198861198872119886
368640 11988711988611988741198861198873119886
18119860 737280 11988731198861198873
20119860737280 119886119887
41198861198872
737280 1198873119886119887119886
20119861737280 119886119887
41198861198874119886
737280 119886119887119886119887119886
24119860737280 119886119887
2
737280 1198873119886
24119861737280 119887
4
737280 119887
28119860737280 119886119887119886119887
31198861198872
737280 119886119887411988611988721198861198872
30119860737280 119887
21198861198874
737280 1198871198861198873
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R H Bing ldquoA homeomorphism between the 3-sphere andthe sum of two solid horned spheresrdquo Annals of MathematicsSecond Series vol 56 pp 354ndash362 1952
[2] D Montgomery and L Zippin ldquoExamples of transformationgroupsrdquo Proceedings of the American Mathematical Society vol5 pp 460ndash465 1954
[3] J W Morgan and H Bass Eds The Smith Conjecture vol 112of Pure and Applied Mathematics Academic Press OrlandoFla USA 1984 Papers presented at the symposium held atColumbia University New York NY USA 1979
[4] C M Edwards and G T Ruttimann ldquoInvolutive and Peircegradings in JBWlowast-triplesrdquo Communications in Algebra vol 31no 6 pp 2819ndash2848 2003
Algebra 15
[5] J Gornicki and B E Rhoades ldquoA general fixed point theoremfor involutionsrdquo Indian Journal of Pure and Applied Mathemat-ics vol 27 no 1 pp 13ndash23 1996
[6] S Montgomery Fixed Rings of Finite Automorphism Groupsof Associative Rings vol 818 of Lecture Notes in MathematicsSpringer Berlin Germany 1980
[7] T A Springer ldquoThe classification of involutions of simplealgebraic groupsrdquo Journal of the Faculty of Science University ofTokyo Section IA Mathematics vol 34 no 3 pp 655ndash670 1987
[8] T A Springer ldquoSome results on algebraic group s with invo-lutionsrdquo in Algebraic Groups and Related Topics (KyotoNagoya1983) vol 6 of Advanced Studies in Pure Mathematics pp 525ndash543 North-Holland Amsterdam The Netherlands 1985
[9] W Feit and J GThompson ldquoSolvability of groups of odd orderrdquoPacific Journal of Mathematics vol 13 pp 775ndash1029 1963
[10] R Brauer and K A Fowler ldquoOn groups of even orderrdquo Annalsof Mathematics vol 62 no 3 pp 565ndash583 1955
[11] D Gorenstein The Classification of Finite Ssimple Groups Vol1 Groups of Noncharacteristic 2 Type The University Series inMathematics Plenum Press New York NY USA 1983
[12] D Gorenstein R Lyons and R Solomon The Classification ofthe Finite Simple Groups vol 40 of Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 1994
[13] C Bates and P Rowley ldquoInvolutions in Conwayrsquos largest simplegrouprdquo LMS Journal of Computation andMathematics vol 7 pp337ndash351 2004
[14] P Rowley and P Taylor ldquoPoint-line collinearity graphs of twosporadic minimal parabolic geometriesrdquo Journal of Algebra vol331 pp 304ndash310 2011
[15] P Rowley and P Taylor ldquoInvolutions in Jankorsquos simple group 1198694rdquoLMS Journal of Computation and Mathematics vol 14 pp 238ndash253 2011
[16] P Taylor ldquoInvolutions in Fischerrsquos sporadic groupsrdquo preprinthttpeprintsmamanacuk1622
[17] The GAP Group GAP-Groups Algorithms Programming Ver-sion 43 2002 httpwwwgap-systemorg
[18] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I The user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997
[19] R Wilson P Walsh J Tripp et al ldquoAtlas o f finite grouprepresentationsrdquo httpbrauermathsqmulacukAtlasv3
[20] J H Conway R T Curtis S P Norton R A Parker and R AWilson Atlas of Finite Groups Clarendon Oxford UK 1985
[21] R AWilson ldquoStandard generators for sporadic simple groupsrdquoJournal of Algebra vol 184 no 2 pp 505ndash515 1996
[22] D Gorenstein Finite Groups Chelsea Publishing New YorkNY USA 2nd edition 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Algebra 11
Table 35 Continued
119883119862
|O| 119892
6119864 15840 119886119887119886119887119886119887119886119887211988611988711988611988721198861198872119886119887
7119860 47520 1198871198861198872119886119887119886
811986147520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198872119886119887119886119887119886119887119886119887119886
119886119887119886119887119886119887
911986095040 119887119886119887119886119887119886119887
31680 119887119886119887119886119887211988611988721198861198871198861198872119886
911986195040 119887
2119886119887119886119887119886
31680 1198861198872119886119887119886119887119886119887119886119887119886119887119886
10119860 95040 119886119887
1011986147520 119887119886119887119886119887
21198861198872119886
47520 11988711988611988721198861198872119886119887119886
11119860 190080 119886
12119860 47520 1198872119886119887119886
12119861
47520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198871198861198872119886119887119886119887119886119887
2
119886119887119886119887119886119887119886119887119886
15840
15840
1003816100381610038161003816100381610038161003816100381610038161003816
11988721198861198872119886119887119886119887
119887119886119887119886119887119886
1211986247520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
1198861198872119886119887119886
1198861198871198861198871198861198871198861198871198861198871198861198872
12119864
47520 11988721198861198872119886
47520 119886119887119886119887
47520 119887119886119887119886119887119886119887119886119887
47520 1198872119886119887119886119887119886119887119886
1411986047520
47520
1003816100381610038161003816100381610038161003816100381610038161003816
11988611988721198861198871198861198872119886119887119886119887
119887
15119862 190080 119886119887119886119887119886
1811986095040 119887119886119887
2119886119887119886119887119886
95040 11988611988721198861198872119886119887119886119887
1811986195040 119887
21198861198872119886119887119886119887119886
95040 1198871198861198871198861198871198861198872119886
2011986095040
95040
1003816100381610038161003816100381610038161003816100381610038161003816
1198871198861198871198861198872119886119887119886
119887119886
Table 36
119883119862
|O| 119892
2119860 630 119886119887211988621198872119886
3119861 8960 11988611988721198862
3119862 1920 11988621198871198862
Table 37
119883119862
|O| 119892
4119861 30240 1198861198872119886
5119861 48384 1198861198871198862
6119862 80640 119886
Table 38
119883119862
|O| 119892
2119860 495 119887119886119887119886211988711988621198871198861198872119886119887
2119861396 119886119887
311988611988721198862119887119886
396 11988621198873119886119887211988621198871198862
31198601320 119887119886
211988711988621198872119886119887
440 11988611988721198861198871198861198872119886119887119886119887
31198612640 119886
211988721198861198871198861198873119886
1584 119886119887119886119887119886211988731198862
3119862 7920 11988711988611988721198861198872119886119887
4119860 2970 1198871198861198871198861198871198861198872119886119887119886
4119861
11880 1198862119887
5940 1198872
2970 1198871198861198871198861198872119886119887
51198609504 119887
21198861198871198862
9504 119887119886211988731198862119887119886
5119861 47520 11988621198872119886
611986011880 119887119886
211988721198861198872119886
3960 119886119887211988621198871198861198871198862119887119886
611986115840 119886
2
15840 119886
6119862
23760 1198871198861198871198861198873119886
23760 11988711988621198872119886119887119886
15840 1198871198861198873119886119887119886
611986423760 119887
2119886
23760 11988621198872
711986095040 119887119886
2
31680 1198871198861198872119886119887119886
811986023760 119887119886119887119886119887
2119886
23760 1198872119886119887119886119887119886119887
811986123760 119886119887119886119887
2119886
23760 11988721198862119887119886
3142 119883 = 2119861 119905 = 119886 See Table 33
3143 119883 = 2119862 119905 = 119888 See Table 34
3144 119883 = 2119863 119905 = (119888(119888119889)12)11 See Table 35
315 119870 cong 1198621199003
3151 119883 = 2119860 119905 = 1198872 See Tables 36 and 37
12 Algebra
Table 39
119883119862
|O| 119892
8119862
23760 11988611988721198861198872
23760 1198872119886211988721198862
23760 1198872119886119887119886
23760 11988711988621198871198861198873
911986031680 119887119886119887
21198861198873
31680 119886119887119886119887119886119887119886119887
9119861 63360 1198861198871198861198871198861198872
1011986047520 119886119887119886
21198872
47520 1198871198861198871198861198872
1011986147520 119886119887119886
47520 119886211988731198862
1211986023760 119886
21198871198862
23760 1198861198873119886
12119861
47520 119887
47520 1198861198871198861198871198862
47520 1198871198861198872119886119887
1411986095040 119887119886119887119886
95040 1198861198872119886119887119886
1511986095040 119886119887
2
95040 119886119887119886119887
15119861 190080 1198861198871198861198872
1811986095040 119886119887
2119886119887
95040 119887119886119887119886119887
21119860 190080 11988711988621198871198862
2411986095040 119886119887
2119886
95040 119886211988721198862
2411986195040 119886
2119887119886
95040 119887119886
3011986095040 119886119887119886
2
95040 11988611988731198862
3152 119883 = 2119861 119905 = ((1198862119887)3119886119887)5 See Tables 38 and 39
316 119870 cong 1198621199002
3161 119883 = 2119860 119905 = 119886 See Table 40
3162 119883 = 2119861 119905 = 119886(119886119887119886119887119886)2119886(119886119887119886119887119886)7 See Table 41
3163 119883 = 2119862 119905 = (119887minus1(1198861198872)2119886119887minus1(1198861198872)4)5 See Table 42
Table 40
119883119862
|O| 119892
2119861 1260 1198871198861198872119886119887
2119862 1008 11988731198861198871198861198873
3119861 14336 119887119886119887
4119862 40320 119887
Table 41
119883119862
|O| 119892
2119860 15 (1198874119886)2119887311988611988721198861198874(119887119886119887)
2119886(119887119886119887)
711988611988711988611988731198861198872(119886119887)2
21198611680 119887
31198861198871198861198873
210 11988741198861198874119886119887119886119887
2119862 2520 (119886119887)5119886(119886119887)
2119886(119886119887)
3
3119861 35840 1198872119886119887
4119860 1920 11988711988611988731198861198872119886119887119886119887
4119862 20160 1198871198861198872119886119887
4119864 161280 119887119886119887
411986513440 119887119886119887119886119887119886119887119886119887
13440 119887311988611988711988611988721198861198872
5119861 344064 1198872
6119864 430080 119887
Table 42
119883119862
|O| 119892
2119860240 119886119887
4119886119887211988611988711988611988741198861198873119886119887311988611988741198861198873119886119887119886
45 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887411988611988741198861198872119886
2119861
720 1198872119886
240 11988741198861198874119886119887311988611988741198861198874119886119887211988611988731198861198872119886
180 119886119887211988611988731198861198872119886119887119886119887311988611988731198861198872
30 1198861198871198861198874119886
2119862
1440 1198861198871198861198873119886119887311988611988721198861198871198861198872119886
1440 119886119887311988611988741198861198873119886119887211988611988721198861198874119886
1440 1198872119886119887119886119887119886119887
311988611988741198861198873
1152 11988611988711988611988731198861198873119886119887311988611988741198861198872119886
360 11988611988711988611988711988611988731198861198872119886119887119886119887311988611988731198861198872
3119860 10240 1198861198874119886119887211988611988711988611988721198861198871198861198872119886
3119861
15360 11988611988731198861198872119886
7680 11988731198861198872
1024 1198861198872119886119887211988611988731198861198873119886
41198603840 119887
411988611988741198861198873119886119887411988611988741198861198874
1440 1198861198871198861198872119886119887211988611988741198861198872
4119861
11520 1198871198861198874119886119887411988611988731198861198873119886
5760 119886119887211988611988711988611988721198861198871198861198873119886
5760 11988611988721198861198874119886119887311988611988741198861198873119886
1440 11988611988741198861198871198861198872119886119887411988611988721198861198871198861198873119886
1440 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887119886
720 119886119887119886119887411988611988721198861198872119886119887211988611988731198861198873
720 1198872119886119887411988611988721198861198872119886119887411988611988731198861198874119886
Algebra 13
Table 42 Continued
119883119862
|O| 119892
4119862
11520 11988611988721198861198873119886
11520 1198861198873119886119887119886119887311988611988721198861198872
5760 119887311988611988741198861198872119886
5760 11988611988731198861198871198861198872
2880 11988611988711988611988741198861198872119886
1440 119887411988611988741198861198873119886119887211988611988721198861198873119886
1440 1198861198872119886119887311988611988731198861198872119886119887119886119887
720 11988611988711988611988711988611988711988611988721198861198871198861198873119886
720 119886119887311988611988741198861198872119886
4119863
23040 1198871198861198873119886119887119886119887211988611988731198861198872
11520 119886119887119886119887311988611988731198861198871198861198873
5760 119886119887411988611988711988611988741198861198872119886
2880 11988611988711988611988711988611988741198861198872119886119887411988611988741198861198873119886
2880 1198861198872119886119887119886119887119886119887
311988611988711988611988741198861198874119886
4119864
23040 1198871198861198871198861198871198861198872
23040 119886119887311988611988721198861198872
11520 1198861198873119886119887211988611988741198861198872
11520 119887311988611988711988611988731198861198872119886
5760 11988731198861198874119886
5760 1198861198871198861198872
5760 1198874119886119887311988611988721198861198871198861198873119886
4119865
23040 119886119887119886119887211988611988721198861198872
23040 1198873119886119887311988611988731198861198874119886
11520 119887411988611988741198861198872119886119887411988611988731198861198872119886
11520 119886119887311988611988741198861198873119886119887211988611988721198861198872
5760 1198872119886119887411988611988721198861198872119886119887211988611988731198861198872119886
5760 1198872119886119887211988611988731198861198873119886119887411988611988721198861198873119886
5760 1198861198873119886119887211988611988741198861198872119886119887311988611988731198861198872119886
5760 1198861198874119886119887119886119887119886119887
3119886119887211988611988721198861198873
5760 119886119887211988611988731198861198871198861198872119886119887211988611988731198861198873
5760 1198874119886119887411988611988721198861198872119886119887211988611988731198861198872119886
2880 119887311988611988721198861198871198861198872
2880 1198873119886119887411988611988731198861198872
1920 11988611988711988611988741198861198872119886119887211988611988741198861198873
1920 1198871198861198873119886119887311988611988711988611988721198861198874
4119866
46080 1198873119886119887311988611988721198861198873119886
46080 1198861198872119886119887311988611988721198861198872
23040 1198861198871198861198872119886119887211988611988731198861198872119886
23040 11988611988731198861198872119886119887311988611988731198861198874119886
5119860 73728 119886119887211988611988731198861198871198861198872119886
5119861184320 119887
21198861198872119886
122880 119887211988611988731198861198872
Table 42 Continued
119883119862
|O| 119892
6119860 92160 1198871198861198873119886119887411988611988731198861198872119886
6119861 122880 119887211988611988741198861198872119886
6119862
46080 1198872119886119887311988611988731198861198872119886
46080 119887311988611988711988611988721198861198872
15360 1198874119886119887211988611988731198861198873119886
15360 119886119887211988611988721198861198873119886119887
6119863
92160 119887411988611988711988611988721198861198874119886
23040 11988741198861198873119886119887411988611988721198861198872
15360 11988611988721198861198872119886119887119886
15360 119886119887411988611988731198861198873119886
6119864
92160 1198871198861198871198861198872
92160 1198871198861198874119886
46080 1198873119886119887211988611988731198861198872119886
46080 11988711988611988721198861198873119886
46080 11988721198861198871198861198874119886
46080 11988611988711988611988741198861198873
15360 119886119887211988611988711988611988741198861198871198861198872119886
6119865
92160 1198861198873119886119887119886
92160 11988611988741198861198872119886
46080 119887311988611988721198861198871198861198872119886
46080 1198861198874119886119887411988611988731198861198873
46080 1198861198873119886119887411988611988731198861198872
46080 11988721198861198872119886119887119886119887119886
7119860 737280 11988741198861198872
811986092160 119887
2119886119887119886
92160 11988611988741198861198873
8119861
92160 119886119887411988611988721198861198872119886119887119886
92160 119886119887411988611988731198861198873119886119887119886
46080 11988611988711988611988731198861198872119886
46080 119886119887311988611988721198861198874119886
23040 119887119886119887311988611988731198861198871198861198873
23040 11988721198861198874119886119887211988611988721198861198874
8119862
92160 1198873119886119887311988611988731198861198874
92160 1198872
92160 1198873
92160 119887119886119887211988611988721198861198872
8119863
92160 119887311988611988731198861198872
92160 11988611988721198861198873119886119887311988611988741198861198872
92160 119887411988611988731198861198873119886
92160 11988711988611988721198861198871198861198873119886
92160 119887311988611988721198861198872
14 Algebra
Table 42 Continued
119883119862
|O| 119892
8119864
92160 1198861198873119886
92160 1198861198872119886
46080 119886119887311988611988711988611988741198861198872
46080 119887311988611988711988611988741198861198872119886
46080 119886119887211988611988721198861198871198861198873
46080 1198872119886119887119886119887119886
46080 1198872119886119887411988611988731198861198873119886
46080 1198872119886119887211988611988731198861198873119886
46080 1198861198872119886119887211988611988731198861198873
46080 119886119887411988611988741198861198873
23040 1198861198874119886119887211988611988721198861198871198861198873
23040 11988741198861198874119886119887211988611988741198861198873
8119865
184320 11988611988741198861198871198861198873119886
184320 11988611988721198861198874119886119887119886
92160 119886119887119886119887311988611988711988611988721198861198872
92160 11988731198861198874119886119887411988611988731198861198872119886
9119860737280 119886119887
41198861198871198861198872
245760 119887211988611988741198861198872
10119860 368640 119887211988611988731198861198872119886
10119861368640 119886119887
21198861198872
368640 11988731198861198873119886
10119862368640 119886119887
4
368640 11988611988711988611988731198861198871198861198873119886
11119860 1474560 11988721198861198873119886
12119860122880 119887119886119887
21198861198871198861198874119886
122880 119887411988611988711988611988741198861198873
12119861368640 119887
211988611988711988611988731198861198872
184320 1198861198874119886119887211988611988741198861198872
12119862368640 119886119887
2119886119887211988611988731198861198872
184320 119886119887411988611988711988611988721198861198872
12119863
184320 119887411988611988711988611988721198861198872
184320 11988611988711988611988711988611988731198861198873119886
92160 119886119887411988611988721198861198873119886
92160 11988611988721198861198873119886119887119886
12119864368640 119887
311988611988721198861198874
368640 11988711988611988731198861198872
12119865
184320 11988721198861198871198861198872119886
184320 11988611988731198861198874119886
184320 11988611988711988611988731198861198873
184320 1198861198871198861198872119886
92160 1198861198872119886119887119886
92160 11988611988741198861198873119886
12119866 368640 11988711988611988741198861198872119886
Table 42 Continued
119883119862
|O| 119892
12119867
184320 1198874119886119887311988611988731198861198872
184320 119886119887311988611988721198861198873119886
184320 11988611988711988611988711988611988741198861198872119886
184320 1198861198874119886119887311988611988741198861198872119886
184320 119887311988611988721198861198872119886119887
184320 119886119887211988611988731198861198872119886
14119860 737280 1198861198874119886
15119860737280 119887
4119886
737280 1198871198861198872119886
16119860
368640 11988611988731198861198872
368640 11988611988721198861198871198861198874
368640 11988731198861198872119886
368640 11988711988611988741198861198873119886
18119860 737280 11988731198861198873
20119860737280 119886119887
41198861198872
737280 1198873119886119887119886
20119861737280 119886119887
41198861198874119886
737280 119886119887119886119887119886
24119860737280 119886119887
2
737280 1198873119886
24119861737280 119887
4
737280 119887
28119860737280 119886119887119886119887
31198861198872
737280 119886119887411988611988721198861198872
30119860737280 119887
21198861198874
737280 1198871198861198873
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R H Bing ldquoA homeomorphism between the 3-sphere andthe sum of two solid horned spheresrdquo Annals of MathematicsSecond Series vol 56 pp 354ndash362 1952
[2] D Montgomery and L Zippin ldquoExamples of transformationgroupsrdquo Proceedings of the American Mathematical Society vol5 pp 460ndash465 1954
[3] J W Morgan and H Bass Eds The Smith Conjecture vol 112of Pure and Applied Mathematics Academic Press OrlandoFla USA 1984 Papers presented at the symposium held atColumbia University New York NY USA 1979
[4] C M Edwards and G T Ruttimann ldquoInvolutive and Peircegradings in JBWlowast-triplesrdquo Communications in Algebra vol 31no 6 pp 2819ndash2848 2003
Algebra 15
[5] J Gornicki and B E Rhoades ldquoA general fixed point theoremfor involutionsrdquo Indian Journal of Pure and Applied Mathemat-ics vol 27 no 1 pp 13ndash23 1996
[6] S Montgomery Fixed Rings of Finite Automorphism Groupsof Associative Rings vol 818 of Lecture Notes in MathematicsSpringer Berlin Germany 1980
[7] T A Springer ldquoThe classification of involutions of simplealgebraic groupsrdquo Journal of the Faculty of Science University ofTokyo Section IA Mathematics vol 34 no 3 pp 655ndash670 1987
[8] T A Springer ldquoSome results on algebraic group s with invo-lutionsrdquo in Algebraic Groups and Related Topics (KyotoNagoya1983) vol 6 of Advanced Studies in Pure Mathematics pp 525ndash543 North-Holland Amsterdam The Netherlands 1985
[9] W Feit and J GThompson ldquoSolvability of groups of odd orderrdquoPacific Journal of Mathematics vol 13 pp 775ndash1029 1963
[10] R Brauer and K A Fowler ldquoOn groups of even orderrdquo Annalsof Mathematics vol 62 no 3 pp 565ndash583 1955
[11] D Gorenstein The Classification of Finite Ssimple Groups Vol1 Groups of Noncharacteristic 2 Type The University Series inMathematics Plenum Press New York NY USA 1983
[12] D Gorenstein R Lyons and R Solomon The Classification ofthe Finite Simple Groups vol 40 of Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 1994
[13] C Bates and P Rowley ldquoInvolutions in Conwayrsquos largest simplegrouprdquo LMS Journal of Computation andMathematics vol 7 pp337ndash351 2004
[14] P Rowley and P Taylor ldquoPoint-line collinearity graphs of twosporadic minimal parabolic geometriesrdquo Journal of Algebra vol331 pp 304ndash310 2011
[15] P Rowley and P Taylor ldquoInvolutions in Jankorsquos simple group 1198694rdquoLMS Journal of Computation and Mathematics vol 14 pp 238ndash253 2011
[16] P Taylor ldquoInvolutions in Fischerrsquos sporadic groupsrdquo preprinthttpeprintsmamanacuk1622
[17] The GAP Group GAP-Groups Algorithms Programming Ver-sion 43 2002 httpwwwgap-systemorg
[18] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I The user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997
[19] R Wilson P Walsh J Tripp et al ldquoAtlas o f finite grouprepresentationsrdquo httpbrauermathsqmulacukAtlasv3
[20] J H Conway R T Curtis S P Norton R A Parker and R AWilson Atlas of Finite Groups Clarendon Oxford UK 1985
[21] R AWilson ldquoStandard generators for sporadic simple groupsrdquoJournal of Algebra vol 184 no 2 pp 505ndash515 1996
[22] D Gorenstein Finite Groups Chelsea Publishing New YorkNY USA 2nd edition 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
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Applied MathematicsJournal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Algebra
Table 39
119883119862
|O| 119892
8119862
23760 11988611988721198861198872
23760 1198872119886211988721198862
23760 1198872119886119887119886
23760 11988711988621198871198861198873
911986031680 119887119886119887
21198861198873
31680 119886119887119886119887119886119887119886119887
9119861 63360 1198861198871198861198871198861198872
1011986047520 119886119887119886
21198872
47520 1198871198861198871198861198872
1011986147520 119886119887119886
47520 119886211988731198862
1211986023760 119886
21198871198862
23760 1198861198873119886
12119861
47520 119887
47520 1198861198871198861198871198862
47520 1198871198861198872119886119887
1411986095040 119887119886119887119886
95040 1198861198872119886119887119886
1511986095040 119886119887
2
95040 119886119887119886119887
15119861 190080 1198861198871198861198872
1811986095040 119886119887
2119886119887
95040 119887119886119887119886119887
21119860 190080 11988711988621198871198862
2411986095040 119886119887
2119886
95040 119886211988721198862
2411986195040 119886
2119887119886
95040 119887119886
3011986095040 119886119887119886
2
95040 11988611988731198862
3152 119883 = 2119861 119905 = ((1198862119887)3119886119887)5 See Tables 38 and 39
316 119870 cong 1198621199002
3161 119883 = 2119860 119905 = 119886 See Table 40
3162 119883 = 2119861 119905 = 119886(119886119887119886119887119886)2119886(119886119887119886119887119886)7 See Table 41
3163 119883 = 2119862 119905 = (119887minus1(1198861198872)2119886119887minus1(1198861198872)4)5 See Table 42
Table 40
119883119862
|O| 119892
2119861 1260 1198871198861198872119886119887
2119862 1008 11988731198861198871198861198873
3119861 14336 119887119886119887
4119862 40320 119887
Table 41
119883119862
|O| 119892
2119860 15 (1198874119886)2119887311988611988721198861198874(119887119886119887)
2119886(119887119886119887)
711988611988711988611988731198861198872(119886119887)2
21198611680 119887
31198861198871198861198873
210 11988741198861198874119886119887119886119887
2119862 2520 (119886119887)5119886(119886119887)
2119886(119886119887)
3
3119861 35840 1198872119886119887
4119860 1920 11988711988611988731198861198872119886119887119886119887
4119862 20160 1198871198861198872119886119887
4119864 161280 119887119886119887
411986513440 119887119886119887119886119887119886119887119886119887
13440 119887311988611988711988611988721198861198872
5119861 344064 1198872
6119864 430080 119887
Table 42
119883119862
|O| 119892
2119860240 119886119887
4119886119887211988611988711988611988741198861198873119886119887311988611988741198861198873119886119887119886
45 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887411988611988741198861198872119886
2119861
720 1198872119886
240 11988741198861198874119886119887311988611988741198861198874119886119887211988611988731198861198872119886
180 119886119887211988611988731198861198872119886119887119886119887311988611988731198861198872
30 1198861198871198861198874119886
2119862
1440 1198861198871198861198873119886119887311988611988721198861198871198861198872119886
1440 119886119887311988611988741198861198873119886119887211988611988721198861198874119886
1440 1198872119886119887119886119887119886119887
311988611988741198861198873
1152 11988611988711988611988731198861198873119886119887311988611988741198861198872119886
360 11988611988711988611988711988611988731198861198872119886119887119886119887311988611988731198861198872
3119860 10240 1198861198874119886119887211988611988711988611988721198861198871198861198872119886
3119861
15360 11988611988731198861198872119886
7680 11988731198861198872
1024 1198861198872119886119887211988611988731198861198873119886
41198603840 119887
411988611988741198861198873119886119887411988611988741198861198874
1440 1198861198871198861198872119886119887211988611988741198861198872
4119861
11520 1198871198861198874119886119887411988611988731198861198873119886
5760 119886119887211988611988711988611988721198861198871198861198873119886
5760 11988611988721198861198874119886119887311988611988741198861198873119886
1440 11988611988741198861198871198861198872119886119887411988611988721198861198871198861198873119886
1440 11988611988721198861198874119886119887311988611988711988611988731198861198874119886119887119886
720 119886119887119886119887411988611988721198861198872119886119887211988611988731198861198873
720 1198872119886119887411988611988721198861198872119886119887411988611988731198861198874119886
Algebra 13
Table 42 Continued
119883119862
|O| 119892
4119862
11520 11988611988721198861198873119886
11520 1198861198873119886119887119886119887311988611988721198861198872
5760 119887311988611988741198861198872119886
5760 11988611988731198861198871198861198872
2880 11988611988711988611988741198861198872119886
1440 119887411988611988741198861198873119886119887211988611988721198861198873119886
1440 1198861198872119886119887311988611988731198861198872119886119887119886119887
720 11988611988711988611988711988611988711988611988721198861198871198861198873119886
720 119886119887311988611988741198861198872119886
4119863
23040 1198871198861198873119886119887119886119887211988611988731198861198872
11520 119886119887119886119887311988611988731198861198871198861198873
5760 119886119887411988611988711988611988741198861198872119886
2880 11988611988711988611988711988611988741198861198872119886119887411988611988741198861198873119886
2880 1198861198872119886119887119886119887119886119887
311988611988711988611988741198861198874119886
4119864
23040 1198871198861198871198861198871198861198872
23040 119886119887311988611988721198861198872
11520 1198861198873119886119887211988611988741198861198872
11520 119887311988611988711988611988731198861198872119886
5760 11988731198861198874119886
5760 1198861198871198861198872
5760 1198874119886119887311988611988721198861198871198861198873119886
4119865
23040 119886119887119886119887211988611988721198861198872
23040 1198873119886119887311988611988731198861198874119886
11520 119887411988611988741198861198872119886119887411988611988731198861198872119886
11520 119886119887311988611988741198861198873119886119887211988611988721198861198872
5760 1198872119886119887411988611988721198861198872119886119887211988611988731198861198872119886
5760 1198872119886119887211988611988731198861198873119886119887411988611988721198861198873119886
5760 1198861198873119886119887211988611988741198861198872119886119887311988611988731198861198872119886
5760 1198861198874119886119887119886119887119886119887
3119886119887211988611988721198861198873
5760 119886119887211988611988731198861198871198861198872119886119887211988611988731198861198873
5760 1198874119886119887411988611988721198861198872119886119887211988611988731198861198872119886
2880 119887311988611988721198861198871198861198872
2880 1198873119886119887411988611988731198861198872
1920 11988611988711988611988741198861198872119886119887211988611988741198861198873
1920 1198871198861198873119886119887311988611988711988611988721198861198874
4119866
46080 1198873119886119887311988611988721198861198873119886
46080 1198861198872119886119887311988611988721198861198872
23040 1198861198871198861198872119886119887211988611988731198861198872119886
23040 11988611988731198861198872119886119887311988611988731198861198874119886
5119860 73728 119886119887211988611988731198861198871198861198872119886
5119861184320 119887
21198861198872119886
122880 119887211988611988731198861198872
Table 42 Continued
119883119862
|O| 119892
6119860 92160 1198871198861198873119886119887411988611988731198861198872119886
6119861 122880 119887211988611988741198861198872119886
6119862
46080 1198872119886119887311988611988731198861198872119886
46080 119887311988611988711988611988721198861198872
15360 1198874119886119887211988611988731198861198873119886
15360 119886119887211988611988721198861198873119886119887
6119863
92160 119887411988611988711988611988721198861198874119886
23040 11988741198861198873119886119887411988611988721198861198872
15360 11988611988721198861198872119886119887119886
15360 119886119887411988611988731198861198873119886
6119864
92160 1198871198861198871198861198872
92160 1198871198861198874119886
46080 1198873119886119887211988611988731198861198872119886
46080 11988711988611988721198861198873119886
46080 11988721198861198871198861198874119886
46080 11988611988711988611988741198861198873
15360 119886119887211988611988711988611988741198861198871198861198872119886
6119865
92160 1198861198873119886119887119886
92160 11988611988741198861198872119886
46080 119887311988611988721198861198871198861198872119886
46080 1198861198874119886119887411988611988731198861198873
46080 1198861198873119886119887411988611988731198861198872
46080 11988721198861198872119886119887119886119887119886
7119860 737280 11988741198861198872
811986092160 119887
2119886119887119886
92160 11988611988741198861198873
8119861
92160 119886119887411988611988721198861198872119886119887119886
92160 119886119887411988611988731198861198873119886119887119886
46080 11988611988711988611988731198861198872119886
46080 119886119887311988611988721198861198874119886
23040 119887119886119887311988611988731198861198871198861198873
23040 11988721198861198874119886119887211988611988721198861198874
8119862
92160 1198873119886119887311988611988731198861198874
92160 1198872
92160 1198873
92160 119887119886119887211988611988721198861198872
8119863
92160 119887311988611988731198861198872
92160 11988611988721198861198873119886119887311988611988741198861198872
92160 119887411988611988731198861198873119886
92160 11988711988611988721198861198871198861198873119886
92160 119887311988611988721198861198872
14 Algebra
Table 42 Continued
119883119862
|O| 119892
8119864
92160 1198861198873119886
92160 1198861198872119886
46080 119886119887311988611988711988611988741198861198872
46080 119887311988611988711988611988741198861198872119886
46080 119886119887211988611988721198861198871198861198873
46080 1198872119886119887119886119887119886
46080 1198872119886119887411988611988731198861198873119886
46080 1198872119886119887211988611988731198861198873119886
46080 1198861198872119886119887211988611988731198861198873
46080 119886119887411988611988741198861198873
23040 1198861198874119886119887211988611988721198861198871198861198873
23040 11988741198861198874119886119887211988611988741198861198873
8119865
184320 11988611988741198861198871198861198873119886
184320 11988611988721198861198874119886119887119886
92160 119886119887119886119887311988611988711988611988721198861198872
92160 11988731198861198874119886119887411988611988731198861198872119886
9119860737280 119886119887
41198861198871198861198872
245760 119887211988611988741198861198872
10119860 368640 119887211988611988731198861198872119886
10119861368640 119886119887
21198861198872
368640 11988731198861198873119886
10119862368640 119886119887
4
368640 11988611988711988611988731198861198871198861198873119886
11119860 1474560 11988721198861198873119886
12119860122880 119887119886119887
21198861198871198861198874119886
122880 119887411988611988711988611988741198861198873
12119861368640 119887
211988611988711988611988731198861198872
184320 1198861198874119886119887211988611988741198861198872
12119862368640 119886119887
2119886119887211988611988731198861198872
184320 119886119887411988611988711988611988721198861198872
12119863
184320 119887411988611988711988611988721198861198872
184320 11988611988711988611988711988611988731198861198873119886
92160 119886119887411988611988721198861198873119886
92160 11988611988721198861198873119886119887119886
12119864368640 119887
311988611988721198861198874
368640 11988711988611988731198861198872
12119865
184320 11988721198861198871198861198872119886
184320 11988611988731198861198874119886
184320 11988611988711988611988731198861198873
184320 1198861198871198861198872119886
92160 1198861198872119886119887119886
92160 11988611988741198861198873119886
12119866 368640 11988711988611988741198861198872119886
Table 42 Continued
119883119862
|O| 119892
12119867
184320 1198874119886119887311988611988731198861198872
184320 119886119887311988611988721198861198873119886
184320 11988611988711988611988711988611988741198861198872119886
184320 1198861198874119886119887311988611988741198861198872119886
184320 119887311988611988721198861198872119886119887
184320 119886119887211988611988731198861198872119886
14119860 737280 1198861198874119886
15119860737280 119887
4119886
737280 1198871198861198872119886
16119860
368640 11988611988731198861198872
368640 11988611988721198861198871198861198874
368640 11988731198861198872119886
368640 11988711988611988741198861198873119886
18119860 737280 11988731198861198873
20119860737280 119886119887
41198861198872
737280 1198873119886119887119886
20119861737280 119886119887
41198861198874119886
737280 119886119887119886119887119886
24119860737280 119886119887
2
737280 1198873119886
24119861737280 119887
4
737280 119887
28119860737280 119886119887119886119887
31198861198872
737280 119886119887411988611988721198861198872
30119860737280 119887
21198861198874
737280 1198871198861198873
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R H Bing ldquoA homeomorphism between the 3-sphere andthe sum of two solid horned spheresrdquo Annals of MathematicsSecond Series vol 56 pp 354ndash362 1952
[2] D Montgomery and L Zippin ldquoExamples of transformationgroupsrdquo Proceedings of the American Mathematical Society vol5 pp 460ndash465 1954
[3] J W Morgan and H Bass Eds The Smith Conjecture vol 112of Pure and Applied Mathematics Academic Press OrlandoFla USA 1984 Papers presented at the symposium held atColumbia University New York NY USA 1979
[4] C M Edwards and G T Ruttimann ldquoInvolutive and Peircegradings in JBWlowast-triplesrdquo Communications in Algebra vol 31no 6 pp 2819ndash2848 2003
Algebra 15
[5] J Gornicki and B E Rhoades ldquoA general fixed point theoremfor involutionsrdquo Indian Journal of Pure and Applied Mathemat-ics vol 27 no 1 pp 13ndash23 1996
[6] S Montgomery Fixed Rings of Finite Automorphism Groupsof Associative Rings vol 818 of Lecture Notes in MathematicsSpringer Berlin Germany 1980
[7] T A Springer ldquoThe classification of involutions of simplealgebraic groupsrdquo Journal of the Faculty of Science University ofTokyo Section IA Mathematics vol 34 no 3 pp 655ndash670 1987
[8] T A Springer ldquoSome results on algebraic group s with invo-lutionsrdquo in Algebraic Groups and Related Topics (KyotoNagoya1983) vol 6 of Advanced Studies in Pure Mathematics pp 525ndash543 North-Holland Amsterdam The Netherlands 1985
[9] W Feit and J GThompson ldquoSolvability of groups of odd orderrdquoPacific Journal of Mathematics vol 13 pp 775ndash1029 1963
[10] R Brauer and K A Fowler ldquoOn groups of even orderrdquo Annalsof Mathematics vol 62 no 3 pp 565ndash583 1955
[11] D Gorenstein The Classification of Finite Ssimple Groups Vol1 Groups of Noncharacteristic 2 Type The University Series inMathematics Plenum Press New York NY USA 1983
[12] D Gorenstein R Lyons and R Solomon The Classification ofthe Finite Simple Groups vol 40 of Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 1994
[13] C Bates and P Rowley ldquoInvolutions in Conwayrsquos largest simplegrouprdquo LMS Journal of Computation andMathematics vol 7 pp337ndash351 2004
[14] P Rowley and P Taylor ldquoPoint-line collinearity graphs of twosporadic minimal parabolic geometriesrdquo Journal of Algebra vol331 pp 304ndash310 2011
[15] P Rowley and P Taylor ldquoInvolutions in Jankorsquos simple group 1198694rdquoLMS Journal of Computation and Mathematics vol 14 pp 238ndash253 2011
[16] P Taylor ldquoInvolutions in Fischerrsquos sporadic groupsrdquo preprinthttpeprintsmamanacuk1622
[17] The GAP Group GAP-Groups Algorithms Programming Ver-sion 43 2002 httpwwwgap-systemorg
[18] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I The user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997
[19] R Wilson P Walsh J Tripp et al ldquoAtlas o f finite grouprepresentationsrdquo httpbrauermathsqmulacukAtlasv3
[20] J H Conway R T Curtis S P Norton R A Parker and R AWilson Atlas of Finite Groups Clarendon Oxford UK 1985
[21] R AWilson ldquoStandard generators for sporadic simple groupsrdquoJournal of Algebra vol 184 no 2 pp 505ndash515 1996
[22] D Gorenstein Finite Groups Chelsea Publishing New YorkNY USA 2nd edition 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Algebra 13
Table 42 Continued
119883119862
|O| 119892
4119862
11520 11988611988721198861198873119886
11520 1198861198873119886119887119886119887311988611988721198861198872
5760 119887311988611988741198861198872119886
5760 11988611988731198861198871198861198872
2880 11988611988711988611988741198861198872119886
1440 119887411988611988741198861198873119886119887211988611988721198861198873119886
1440 1198861198872119886119887311988611988731198861198872119886119887119886119887
720 11988611988711988611988711988611988711988611988721198861198871198861198873119886
720 119886119887311988611988741198861198872119886
4119863
23040 1198871198861198873119886119887119886119887211988611988731198861198872
11520 119886119887119886119887311988611988731198861198871198861198873
5760 119886119887411988611988711988611988741198861198872119886
2880 11988611988711988611988711988611988741198861198872119886119887411988611988741198861198873119886
2880 1198861198872119886119887119886119887119886119887
311988611988711988611988741198861198874119886
4119864
23040 1198871198861198871198861198871198861198872
23040 119886119887311988611988721198861198872
11520 1198861198873119886119887211988611988741198861198872
11520 119887311988611988711988611988731198861198872119886
5760 11988731198861198874119886
5760 1198861198871198861198872
5760 1198874119886119887311988611988721198861198871198861198873119886
4119865
23040 119886119887119886119887211988611988721198861198872
23040 1198873119886119887311988611988731198861198874119886
11520 119887411988611988741198861198872119886119887411988611988731198861198872119886
11520 119886119887311988611988741198861198873119886119887211988611988721198861198872
5760 1198872119886119887411988611988721198861198872119886119887211988611988731198861198872119886
5760 1198872119886119887211988611988731198861198873119886119887411988611988721198861198873119886
5760 1198861198873119886119887211988611988741198861198872119886119887311988611988731198861198872119886
5760 1198861198874119886119887119886119887119886119887
3119886119887211988611988721198861198873
5760 119886119887211988611988731198861198871198861198872119886119887211988611988731198861198873
5760 1198874119886119887411988611988721198861198872119886119887211988611988731198861198872119886
2880 119887311988611988721198861198871198861198872
2880 1198873119886119887411988611988731198861198872
1920 11988611988711988611988741198861198872119886119887211988611988741198861198873
1920 1198871198861198873119886119887311988611988711988611988721198861198874
4119866
46080 1198873119886119887311988611988721198861198873119886
46080 1198861198872119886119887311988611988721198861198872
23040 1198861198871198861198872119886119887211988611988731198861198872119886
23040 11988611988731198861198872119886119887311988611988731198861198874119886
5119860 73728 119886119887211988611988731198861198871198861198872119886
5119861184320 119887
21198861198872119886
122880 119887211988611988731198861198872
Table 42 Continued
119883119862
|O| 119892
6119860 92160 1198871198861198873119886119887411988611988731198861198872119886
6119861 122880 119887211988611988741198861198872119886
6119862
46080 1198872119886119887311988611988731198861198872119886
46080 119887311988611988711988611988721198861198872
15360 1198874119886119887211988611988731198861198873119886
15360 119886119887211988611988721198861198873119886119887
6119863
92160 119887411988611988711988611988721198861198874119886
23040 11988741198861198873119886119887411988611988721198861198872
15360 11988611988721198861198872119886119887119886
15360 119886119887411988611988731198861198873119886
6119864
92160 1198871198861198871198861198872
92160 1198871198861198874119886
46080 1198873119886119887211988611988731198861198872119886
46080 11988711988611988721198861198873119886
46080 11988721198861198871198861198874119886
46080 11988611988711988611988741198861198873
15360 119886119887211988611988711988611988741198861198871198861198872119886
6119865
92160 1198861198873119886119887119886
92160 11988611988741198861198872119886
46080 119887311988611988721198861198871198861198872119886
46080 1198861198874119886119887411988611988731198861198873
46080 1198861198873119886119887411988611988731198861198872
46080 11988721198861198872119886119887119886119887119886
7119860 737280 11988741198861198872
811986092160 119887
2119886119887119886
92160 11988611988741198861198873
8119861
92160 119886119887411988611988721198861198872119886119887119886
92160 119886119887411988611988731198861198873119886119887119886
46080 11988611988711988611988731198861198872119886
46080 119886119887311988611988721198861198874119886
23040 119887119886119887311988611988731198861198871198861198873
23040 11988721198861198874119886119887211988611988721198861198874
8119862
92160 1198873119886119887311988611988731198861198874
92160 1198872
92160 1198873
92160 119887119886119887211988611988721198861198872
8119863
92160 119887311988611988731198861198872
92160 11988611988721198861198873119886119887311988611988741198861198872
92160 119887411988611988731198861198873119886
92160 11988711988611988721198861198871198861198873119886
92160 119887311988611988721198861198872
14 Algebra
Table 42 Continued
119883119862
|O| 119892
8119864
92160 1198861198873119886
92160 1198861198872119886
46080 119886119887311988611988711988611988741198861198872
46080 119887311988611988711988611988741198861198872119886
46080 119886119887211988611988721198861198871198861198873
46080 1198872119886119887119886119887119886
46080 1198872119886119887411988611988731198861198873119886
46080 1198872119886119887211988611988731198861198873119886
46080 1198861198872119886119887211988611988731198861198873
46080 119886119887411988611988741198861198873
23040 1198861198874119886119887211988611988721198861198871198861198873
23040 11988741198861198874119886119887211988611988741198861198873
8119865
184320 11988611988741198861198871198861198873119886
184320 11988611988721198861198874119886119887119886
92160 119886119887119886119887311988611988711988611988721198861198872
92160 11988731198861198874119886119887411988611988731198861198872119886
9119860737280 119886119887
41198861198871198861198872
245760 119887211988611988741198861198872
10119860 368640 119887211988611988731198861198872119886
10119861368640 119886119887
21198861198872
368640 11988731198861198873119886
10119862368640 119886119887
4
368640 11988611988711988611988731198861198871198861198873119886
11119860 1474560 11988721198861198873119886
12119860122880 119887119886119887
21198861198871198861198874119886
122880 119887411988611988711988611988741198861198873
12119861368640 119887
211988611988711988611988731198861198872
184320 1198861198874119886119887211988611988741198861198872
12119862368640 119886119887
2119886119887211988611988731198861198872
184320 119886119887411988611988711988611988721198861198872
12119863
184320 119887411988611988711988611988721198861198872
184320 11988611988711988611988711988611988731198861198873119886
92160 119886119887411988611988721198861198873119886
92160 11988611988721198861198873119886119887119886
12119864368640 119887
311988611988721198861198874
368640 11988711988611988731198861198872
12119865
184320 11988721198861198871198861198872119886
184320 11988611988731198861198874119886
184320 11988611988711988611988731198861198873
184320 1198861198871198861198872119886
92160 1198861198872119886119887119886
92160 11988611988741198861198873119886
12119866 368640 11988711988611988741198861198872119886
Table 42 Continued
119883119862
|O| 119892
12119867
184320 1198874119886119887311988611988731198861198872
184320 119886119887311988611988721198861198873119886
184320 11988611988711988611988711988611988741198861198872119886
184320 1198861198874119886119887311988611988741198861198872119886
184320 119887311988611988721198861198872119886119887
184320 119886119887211988611988731198861198872119886
14119860 737280 1198861198874119886
15119860737280 119887
4119886
737280 1198871198861198872119886
16119860
368640 11988611988731198861198872
368640 11988611988721198861198871198861198874
368640 11988731198861198872119886
368640 11988711988611988741198861198873119886
18119860 737280 11988731198861198873
20119860737280 119886119887
41198861198872
737280 1198873119886119887119886
20119861737280 119886119887
41198861198874119886
737280 119886119887119886119887119886
24119860737280 119886119887
2
737280 1198873119886
24119861737280 119887
4
737280 119887
28119860737280 119886119887119886119887
31198861198872
737280 119886119887411988611988721198861198872
30119860737280 119887
21198861198874
737280 1198871198861198873
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R H Bing ldquoA homeomorphism between the 3-sphere andthe sum of two solid horned spheresrdquo Annals of MathematicsSecond Series vol 56 pp 354ndash362 1952
[2] D Montgomery and L Zippin ldquoExamples of transformationgroupsrdquo Proceedings of the American Mathematical Society vol5 pp 460ndash465 1954
[3] J W Morgan and H Bass Eds The Smith Conjecture vol 112of Pure and Applied Mathematics Academic Press OrlandoFla USA 1984 Papers presented at the symposium held atColumbia University New York NY USA 1979
[4] C M Edwards and G T Ruttimann ldquoInvolutive and Peircegradings in JBWlowast-triplesrdquo Communications in Algebra vol 31no 6 pp 2819ndash2848 2003
Algebra 15
[5] J Gornicki and B E Rhoades ldquoA general fixed point theoremfor involutionsrdquo Indian Journal of Pure and Applied Mathemat-ics vol 27 no 1 pp 13ndash23 1996
[6] S Montgomery Fixed Rings of Finite Automorphism Groupsof Associative Rings vol 818 of Lecture Notes in MathematicsSpringer Berlin Germany 1980
[7] T A Springer ldquoThe classification of involutions of simplealgebraic groupsrdquo Journal of the Faculty of Science University ofTokyo Section IA Mathematics vol 34 no 3 pp 655ndash670 1987
[8] T A Springer ldquoSome results on algebraic group s with invo-lutionsrdquo in Algebraic Groups and Related Topics (KyotoNagoya1983) vol 6 of Advanced Studies in Pure Mathematics pp 525ndash543 North-Holland Amsterdam The Netherlands 1985
[9] W Feit and J GThompson ldquoSolvability of groups of odd orderrdquoPacific Journal of Mathematics vol 13 pp 775ndash1029 1963
[10] R Brauer and K A Fowler ldquoOn groups of even orderrdquo Annalsof Mathematics vol 62 no 3 pp 565ndash583 1955
[11] D Gorenstein The Classification of Finite Ssimple Groups Vol1 Groups of Noncharacteristic 2 Type The University Series inMathematics Plenum Press New York NY USA 1983
[12] D Gorenstein R Lyons and R Solomon The Classification ofthe Finite Simple Groups vol 40 of Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 1994
[13] C Bates and P Rowley ldquoInvolutions in Conwayrsquos largest simplegrouprdquo LMS Journal of Computation andMathematics vol 7 pp337ndash351 2004
[14] P Rowley and P Taylor ldquoPoint-line collinearity graphs of twosporadic minimal parabolic geometriesrdquo Journal of Algebra vol331 pp 304ndash310 2011
[15] P Rowley and P Taylor ldquoInvolutions in Jankorsquos simple group 1198694rdquoLMS Journal of Computation and Mathematics vol 14 pp 238ndash253 2011
[16] P Taylor ldquoInvolutions in Fischerrsquos sporadic groupsrdquo preprinthttpeprintsmamanacuk1622
[17] The GAP Group GAP-Groups Algorithms Programming Ver-sion 43 2002 httpwwwgap-systemorg
[18] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I The user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997
[19] R Wilson P Walsh J Tripp et al ldquoAtlas o f finite grouprepresentationsrdquo httpbrauermathsqmulacukAtlasv3
[20] J H Conway R T Curtis S P Norton R A Parker and R AWilson Atlas of Finite Groups Clarendon Oxford UK 1985
[21] R AWilson ldquoStandard generators for sporadic simple groupsrdquoJournal of Algebra vol 184 no 2 pp 505ndash515 1996
[22] D Gorenstein Finite Groups Chelsea Publishing New YorkNY USA 2nd edition 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Algebra
Table 42 Continued
119883119862
|O| 119892
8119864
92160 1198861198873119886
92160 1198861198872119886
46080 119886119887311988611988711988611988741198861198872
46080 119887311988611988711988611988741198861198872119886
46080 119886119887211988611988721198861198871198861198873
46080 1198872119886119887119886119887119886
46080 1198872119886119887411988611988731198861198873119886
46080 1198872119886119887211988611988731198861198873119886
46080 1198861198872119886119887211988611988731198861198873
46080 119886119887411988611988741198861198873
23040 1198861198874119886119887211988611988721198861198871198861198873
23040 11988741198861198874119886119887211988611988741198861198873
8119865
184320 11988611988741198861198871198861198873119886
184320 11988611988721198861198874119886119887119886
92160 119886119887119886119887311988611988711988611988721198861198872
92160 11988731198861198874119886119887411988611988731198861198872119886
9119860737280 119886119887
41198861198871198861198872
245760 119887211988611988741198861198872
10119860 368640 119887211988611988731198861198872119886
10119861368640 119886119887
21198861198872
368640 11988731198861198873119886
10119862368640 119886119887
4
368640 11988611988711988611988731198861198871198861198873119886
11119860 1474560 11988721198861198873119886
12119860122880 119887119886119887
21198861198871198861198874119886
122880 119887411988611988711988611988741198861198873
12119861368640 119887
211988611988711988611988731198861198872
184320 1198861198874119886119887211988611988741198861198872
12119862368640 119886119887
2119886119887211988611988731198861198872
184320 119886119887411988611988711988611988721198861198872
12119863
184320 119887411988611988711988611988721198861198872
184320 11988611988711988611988711988611988731198861198873119886
92160 119886119887411988611988721198861198873119886
92160 11988611988721198861198873119886119887119886
12119864368640 119887
311988611988721198861198874
368640 11988711988611988731198861198872
12119865
184320 11988721198861198871198861198872119886
184320 11988611988731198861198874119886
184320 11988611988711988611988731198861198873
184320 1198861198871198861198872119886
92160 1198861198872119886119887119886
92160 11988611988741198861198873119886
12119866 368640 11988711988611988741198861198872119886
Table 42 Continued
119883119862
|O| 119892
12119867
184320 1198874119886119887311988611988731198861198872
184320 119886119887311988611988721198861198873119886
184320 11988611988711988611988711988611988741198861198872119886
184320 1198861198874119886119887311988611988741198861198872119886
184320 119887311988611988721198861198872119886119887
184320 119886119887211988611988731198861198872119886
14119860 737280 1198861198874119886
15119860737280 119887
4119886
737280 1198871198861198872119886
16119860
368640 11988611988731198861198872
368640 11988611988721198861198871198861198874
368640 11988731198861198872119886
368640 11988711988611988741198861198873119886
18119860 737280 11988731198861198873
20119860737280 119886119887
41198861198872
737280 1198873119886119887119886
20119861737280 119886119887
41198861198874119886
737280 119886119887119886119887119886
24119860737280 119886119887
2
737280 1198873119886
24119861737280 119887
4
737280 119887
28119860737280 119886119887119886119887
31198861198872
737280 119886119887411988611988721198861198872
30119860737280 119887
21198861198874
737280 1198871198861198873
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] R H Bing ldquoA homeomorphism between the 3-sphere andthe sum of two solid horned spheresrdquo Annals of MathematicsSecond Series vol 56 pp 354ndash362 1952
[2] D Montgomery and L Zippin ldquoExamples of transformationgroupsrdquo Proceedings of the American Mathematical Society vol5 pp 460ndash465 1954
[3] J W Morgan and H Bass Eds The Smith Conjecture vol 112of Pure and Applied Mathematics Academic Press OrlandoFla USA 1984 Papers presented at the symposium held atColumbia University New York NY USA 1979
[4] C M Edwards and G T Ruttimann ldquoInvolutive and Peircegradings in JBWlowast-triplesrdquo Communications in Algebra vol 31no 6 pp 2819ndash2848 2003
Algebra 15
[5] J Gornicki and B E Rhoades ldquoA general fixed point theoremfor involutionsrdquo Indian Journal of Pure and Applied Mathemat-ics vol 27 no 1 pp 13ndash23 1996
[6] S Montgomery Fixed Rings of Finite Automorphism Groupsof Associative Rings vol 818 of Lecture Notes in MathematicsSpringer Berlin Germany 1980
[7] T A Springer ldquoThe classification of involutions of simplealgebraic groupsrdquo Journal of the Faculty of Science University ofTokyo Section IA Mathematics vol 34 no 3 pp 655ndash670 1987
[8] T A Springer ldquoSome results on algebraic group s with invo-lutionsrdquo in Algebraic Groups and Related Topics (KyotoNagoya1983) vol 6 of Advanced Studies in Pure Mathematics pp 525ndash543 North-Holland Amsterdam The Netherlands 1985
[9] W Feit and J GThompson ldquoSolvability of groups of odd orderrdquoPacific Journal of Mathematics vol 13 pp 775ndash1029 1963
[10] R Brauer and K A Fowler ldquoOn groups of even orderrdquo Annalsof Mathematics vol 62 no 3 pp 565ndash583 1955
[11] D Gorenstein The Classification of Finite Ssimple Groups Vol1 Groups of Noncharacteristic 2 Type The University Series inMathematics Plenum Press New York NY USA 1983
[12] D Gorenstein R Lyons and R Solomon The Classification ofthe Finite Simple Groups vol 40 of Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 1994
[13] C Bates and P Rowley ldquoInvolutions in Conwayrsquos largest simplegrouprdquo LMS Journal of Computation andMathematics vol 7 pp337ndash351 2004
[14] P Rowley and P Taylor ldquoPoint-line collinearity graphs of twosporadic minimal parabolic geometriesrdquo Journal of Algebra vol331 pp 304ndash310 2011
[15] P Rowley and P Taylor ldquoInvolutions in Jankorsquos simple group 1198694rdquoLMS Journal of Computation and Mathematics vol 14 pp 238ndash253 2011
[16] P Taylor ldquoInvolutions in Fischerrsquos sporadic groupsrdquo preprinthttpeprintsmamanacuk1622
[17] The GAP Group GAP-Groups Algorithms Programming Ver-sion 43 2002 httpwwwgap-systemorg
[18] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I The user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997
[19] R Wilson P Walsh J Tripp et al ldquoAtlas o f finite grouprepresentationsrdquo httpbrauermathsqmulacukAtlasv3
[20] J H Conway R T Curtis S P Norton R A Parker and R AWilson Atlas of Finite Groups Clarendon Oxford UK 1985
[21] R AWilson ldquoStandard generators for sporadic simple groupsrdquoJournal of Algebra vol 184 no 2 pp 505ndash515 1996
[22] D Gorenstein Finite Groups Chelsea Publishing New YorkNY USA 2nd edition 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Algebra 15
[5] J Gornicki and B E Rhoades ldquoA general fixed point theoremfor involutionsrdquo Indian Journal of Pure and Applied Mathemat-ics vol 27 no 1 pp 13ndash23 1996
[6] S Montgomery Fixed Rings of Finite Automorphism Groupsof Associative Rings vol 818 of Lecture Notes in MathematicsSpringer Berlin Germany 1980
[7] T A Springer ldquoThe classification of involutions of simplealgebraic groupsrdquo Journal of the Faculty of Science University ofTokyo Section IA Mathematics vol 34 no 3 pp 655ndash670 1987
[8] T A Springer ldquoSome results on algebraic group s with invo-lutionsrdquo in Algebraic Groups and Related Topics (KyotoNagoya1983) vol 6 of Advanced Studies in Pure Mathematics pp 525ndash543 North-Holland Amsterdam The Netherlands 1985
[9] W Feit and J GThompson ldquoSolvability of groups of odd orderrdquoPacific Journal of Mathematics vol 13 pp 775ndash1029 1963
[10] R Brauer and K A Fowler ldquoOn groups of even orderrdquo Annalsof Mathematics vol 62 no 3 pp 565ndash583 1955
[11] D Gorenstein The Classification of Finite Ssimple Groups Vol1 Groups of Noncharacteristic 2 Type The University Series inMathematics Plenum Press New York NY USA 1983
[12] D Gorenstein R Lyons and R Solomon The Classification ofthe Finite Simple Groups vol 40 of Mathematical Surveys andMonographs American Mathematical Society Providence RIUSA 1994
[13] C Bates and P Rowley ldquoInvolutions in Conwayrsquos largest simplegrouprdquo LMS Journal of Computation andMathematics vol 7 pp337ndash351 2004
[14] P Rowley and P Taylor ldquoPoint-line collinearity graphs of twosporadic minimal parabolic geometriesrdquo Journal of Algebra vol331 pp 304ndash310 2011
[15] P Rowley and P Taylor ldquoInvolutions in Jankorsquos simple group 1198694rdquoLMS Journal of Computation and Mathematics vol 14 pp 238ndash253 2011
[16] P Taylor ldquoInvolutions in Fischerrsquos sporadic groupsrdquo preprinthttpeprintsmamanacuk1622
[17] The GAP Group GAP-Groups Algorithms Programming Ver-sion 43 2002 httpwwwgap-systemorg
[18] W Bosma J Cannon and C Playoust ldquoThe Magma algebrasystem I The user languagerdquo Journal of Symbolic Computationvol 24 no 3-4 pp 235ndash265 1997
[19] R Wilson P Walsh J Tripp et al ldquoAtlas o f finite grouprepresentationsrdquo httpbrauermathsqmulacukAtlasv3
[20] J H Conway R T Curtis S P Norton R A Parker and R AWilson Atlas of Finite Groups Clarendon Oxford UK 1985
[21] R AWilson ldquoStandard generators for sporadic simple groupsrdquoJournal of Algebra vol 184 no 2 pp 505ndash515 1996
[22] D Gorenstein Finite Groups Chelsea Publishing New YorkNY USA 2nd edition 1980
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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