Finite subgroups of the group GL(n,Z)

L ~. TId{ATU RI!I C i TED

I. M. I. Kakhro, A. P. Nal'ya, and E. Kh. Tyugu, The PR?TZ [nstr~ment,,] System of Computer Programming [in Russian], Finansy i Statistika, Hosc:~w (1981).

2. S.C. Kleene, introduction to Metamathematics, Van Nostrand, Prim=e ton, N. J. (1952). 3. E.H. Tyugu, "A Progran~ming system with automatic program synthesis.." Lect. Notes Corn-

put. Sci., No. 47, 251-267 (]977). 4. I. Copy, Symbolic l.ogic, Macmillan, New ~fork (1954). 5. D. Prawitz, Natural Deduction, Aguist and Wiksell, Stockholm (1965). 6. N N. Vorob'ev, "A new deducibility algorithm in constructive propositional calculus, ~'

Tr. Mat. Inst. Akad. Nauk SSSR, 52, 193-226 (1958). 7. R.A. Plyushkyavichus, "A version of constructive propositional calculus without struc-

tural deduction rules," Dokl. Akad. Nauk SSSR, 161, No. 2, 292-295 (1965). 8. J. Bi'Dennis, J. B. Fossin, and J. P. Lindermann, "Data flow schemes," Teer. Program-

mirovaniya, 2, 7-43 (1972). 9. N.N. Nepeivoda, "Relationship between natural deduction rules and operators of high-

level algorithmic languages," Dokl. Akado Nauk SSSR, 239, No. 3, 526-529 (1978).

FINITE SUBGROUPS OF THE GROUP GL(n, Z)

P. M. Gudivok, A. A. Kiri]yuk, V. P. Rud'ko, and A. I. Tsitkin

UDC 512.54+519.44

In connection with the attainments of the theory of integer represenEaEions of groups, interest has recently heightened in the study of finite subgroups of the complete linear GL(n, Z) (Z is the ring of rational integers) and their applicatEons in multidimensional crystallography. In 1948, Zassenhaus [i] showed that the description of n-dimensional crys- tallographic groups reduces to the description of finite subgroups of the group GL(n, Z) to the accuracy of conjugacy. Let us note that even at the end of the past century E. S. Fedo- rov and Schoenfliess proved by geometric methods that the group GL(3, Z) contains exactly 73 of the class of conjugate finite subgroups. In 1965 Dade [2] classified all the maximal fi- nite subgroups of the group GL(4, Z) to conjugacy accuracy. By using the idea of G. F. Voron', S. S. Ryshkov [3] proposed an algorithm to describe nonconjugate maximal finite sub- groups of the group GL(n, Z), and showed that the group GL(5, Z) contains exactly 17 such subgroups. In 1972 Broun, Neubuser, and Zassenhaus [4] found all the nonconjuga~e finite subgroups of the group CL(4, Z) by using the Dade [2] results and an electronic computer. There turned out to be 710 such groups. An algorithm to describe all reducible finite sub- groups of the group GL(n, Z) if all the finite subgroups of the group GL(n - I, Z) are known, is mentioned in [4]. The problem of studying irreducible finite subgroups of the group GL(n, Z) thereby became urgent. Plesken and Pohst [5] found the classification algorithm for all Z-representations of the finite group G that are Q-equivalent to a given irreducible Z-repre- sentation of the group G (Q is the field of rational numbers). By using this algorithm and an electronic computer, they described the maximal absolutely irreducible finite subgroups of the group GL(n, Z) (n <~ 9).

D. A. Suprunenko, V. P. Platonov, A. E. Zaleskii, et al. (see [6]) investigated certain classes of periodic subgroups of the group GL(n, K) (K is an arbitrary field). D.A. Supru- nenko and R. T. Vol'vachev (see [6]) described all Sylow p-subgroups of the group GL(n, K) to conjugacy accuracy. D..A. Suprunenko [7] and T. I. Kopylova [8] classified all noncon- jugate finite irreducible solvable subgroups of the group GL(p, Q) (p an odd prime). There results from the V. P. Platonov results [9] that any minimal irreducible subgroup of the group group GL(n, Q) is always finite. Recently Suprunenko [10] established that for n = p such that a group is either solvable or a simple non-Abelian group.

Using the results mentioned about linear groups, Sy]ow q-subgroups of the group GL(n, Z) and solvable irreducible finite subgroups of the group GL(p, Z) (p an odd prime) are investi- gated in this paper. It is shown that for any prime q two Sylow q-subgroups from GL(n, Z)

Translated from Kibernetika, No. 6, pp. 71-82, November-December, 1982. Original ar- ticle submitted June 30, 1982.

788 0011-4235/82/1806-0788507.50 ~D 1983 Plenum Publishing Corporation

are conjugate if and only if n <~ 2. ~Fne iollowing approach is proposed to study the finite solvable irreducible subgroups of the group GL(p, Z) o Initially a relation is established between the finite linear groups and the moduli over group rings. Then the problem of de- scribing the modu!i corresponding to the finite, solvable, irreducible subgroups of the group GL(p, Z) is reduced to the problem of searching for certain systems of polynomials over the field of the two elements. An algorithm is constructed for the search of these systems and its foundation is given. A program is produced according to this algorithm, which is used on the ES-I020 electronic computer to find all such systems of polynomials for primes p < !000o Returning from the system of polynomials to the moduli and the subgroups, sets of mini- mal, irreducible, solvable, subgroups from the group GL(p, Z) can be found such that every minimal irreducible solvable subgroup of the group GL(p, Z) is conjugate to one of those found. In the case when 2 is a primitive or semiprimitive root modulo p, all finite irre- ducible, solvable subgroups of the group GL(p, Z) are classified to conjugaey accuracy by using the algorithm constructed and the Suprunenko results [7]. In the general case, the algorithm constructed affords the possibility of setting the lower bound t of the number of nonconjugate minimal irreducible solvable subgroups of the group GL(p, Z) (p is any odd prime), t = 5.2 s-I + s -- 3, where s = (p -- 1)/m, and m is an exponent to which 2modp belongs. In the case when s = I or 2, this estimate is achieved. Finally, it is established that the group GL(p, Z) contains no less than 3s + I for s > I and no less than 3 for s = I pairwise nonconjugate maximal irreducible finite subgroups.

I. SYLOW p-SUBGROUPS OF THE GROUP GL(n, Z)

Suprunenko and Volvachev (see [6]) described Sylow p-subgroups of the group GL(n, K) with conjugacy accuracy, where K is an arbitrary field. Using these results and the theory of integer representations of finite groups (see [11]), we investigate Sylow p-subgroups of the group GL(n, Z).

LE~MA I (see [6]). An arbitrary periodic subgroup of the group GL(n, Q) is finite.

LEMMA 2 (see [I]]). An arbitrary finite subgroup H of the group GL(n, Q) is conjugate to a certain subgroup H I of the group GL(n, Z).

Definition I. The subgroups H I and H2 of the group GL(n, Z) are called Q-conjugate if there exists a matrix CCGL (n, Q) such that C-IHIC = He.

Definition 2. The Sylow p-subgroup of the group GL(n, Z), being also a Sylow p-subgroup of the group GL(n, Q), is called a Q-Sylow p-subgroup of the group GL(n, Z).

LEMMA 3. For an arbitrary prime number p the Q-Sylow p-subgroups of the group GL(n, Z) are pairwise Q-conjugate.

The proof of the lemma results from the results of Vol'vachev (see [6]).

As Abold and Plesken [12] showed, Sylow p-subgroups of the group GL(n, Z) can exist Which are now Q-Sylow p-subgroups of the group GL(n, Z). By virtue of Lemma 2, every Sylow p-sub- group of the group GL(n, Q) is conjugate to a certain Sylow p-subgroup of the group GL(n, Z).

Let us present the description, needed later, of Q-Sylow p-subgroups of the group GL(n, Z) (n > I) obtained by Vol'vachev to Q-conjugacy accuracy (see [6]).

Let Npr be a Sylow p-subgroup of the symmetric group Spr , H a subgroup of the group

GL(n, Q), H ~ Npr the interlacing of the groups H and Npr, E a primitive complex root of

degree p of I, and g a matrix corresponding to a polynomial dividing a circle ~p(X) of order

Let p z 2 and Pp = <~>. We represent the number n in the form n = (p -- 1)n0 + t (0 ~< t < p -- I), where no = b0 + blp + ... + bspS (bs ~ 0; 0 ~< bj < p; j = 0, I, ...,s). Then any Q-Sylow p-subgroup of the group GL(n, Z) is Q-conjugate to the group

where G diag [1 (~ G~ao o), G(a~ ) (~)

G~ = P , ~ Np~ (r = 0, I . . . . . s),

H (~) = d i a g {H~ . . . . . H ~ ] (Hi = H; i = t . . . . . rn).

789

The group Gpr is an irreducible Q-Sylow p-subgroup of the groupGL((p- 1)p r, Z) (r = 0, I,.oo, s).

Furthermore, let p = 2. We represent the number n in the form

n = a0 + a ~ . 2 § ~.. § a~.2 ~

( a . ~ 1; 0 ~ a j ~ 1; / = 0 , 1 . . . . . s - - l ) .

Any Q-Sylow 2 - s u b g r o u p of t h e g roup GL(n, Z) i s c o n j u g a t e to t he g roup

G = diag [G(/), ~(~ (~0 u21 . . . . . O~s ], where G2o == { q__-__ !},

G~ = H ~ N2~_~ (r = 1 . . . . . s),

H = < ' \ 1 0

The group G2r is an irreducible Q-Sylow 2-subgroup of the group GL(2 r, Z) (r = I, 2,~176176 s).

The description of Q-Sylow p-subgroups of the group GL(n, Z) is thereby reduced to Q- eonjugacy accuracy to a description of Sylow p-subgroups Npr of the group Spr (see []3], say).

The group Npr is of order ph [h = (pr _ 1)/(p -- I)] and is generated by the following r ele- ments of order p:

c d = ~ [ ] , p a + ! . . . . . (p - -1 )pa +7] 1 ~ ] ~ p d

( d = 0 , 1 . . . . . r - - i ) ,

Let us note that the algorithm to find Sylow p-subgroups of a finite group by using an electronic computer is presented by Kennon (see [14]).

LEMiMA 4 (see [6]). For p > 2 the Sylow p-subgroup of the group GL(n, Q) is irreducible if and only if n = (p -- 1)p r (r ~ 0).

LEMMA 5. A prime p > 2 always exists such that the Sylow p-subgroup of the group GL(n, Q) (n > 2) is reducible.

Proof. By virtue of Lemma 4, for odd n > I any Sylow p-subgroup (p > 2) of the group GL(n, Q) is reducible.

Furthermore, let n be even. For n = 4 the Sylow 3-subgroup of the group GL(4, Q) is evidently reducible. Let n = 2m > 4. Then three distinct odd primes Pl, P2, P3 evidently exist such that nontrivial Sylow pi-subgroups are in GL(n, Q). Let Pl < p2 < P~. We assume that for any Pi (i = I, 2, 3) the Sylow Pi-subgroups of the group GL(2m, Q) (m > 2) are ir- reducible. Then by virtue of Lemma 4, 2m = (p -- 1)p~i = (P2 -- I)p~2 (P3 -- I)p~3. A con- tradiction hence follows easily.

The lemma is proved.

LEY~I~ 6 (see [11]). Let G be a finite p-group (p ~ 2). The matrix Z-representation of T

the group G is not decomposable over Z if and only if it is not decomposable over Zp (Z; is a ring of rational p-adic integers).

LEF~4A 7 [15]. Let At and A2 be nonequivalent irreducible matrix Q-representations of the finite p-group G (p ~ 2). Then the group G possesses a nondecomposable Z-representation F of the form

P :g--~ (gCG), A; (g)

T where Ai is a certain irreducible Z-representation of the group G that is Q-equivalent to the representation Ai (i = 1, 2).

THEOREM I. For any prime p the Sylow p-subgroups of the group GL(n, Z) are pairwise conjugate if and only if n ~ 2.

Proof. The sufficiency is evident. We prove the necessity. Let n > 2. By virtue of Lemma 5 a prime number p > 2 exists such that the nontrivial Sylow p-subgroup of the group

790

GL(n, Q) is reducible. It can evidently be considered that G has the form (I). It is easy to see that the groups Gp_~ = I, Gp0, Gpl,...,Gps are not pairwise isomorphic.

Let us consider two eases.

I. Let there be i and j (--I ~< i < j ~< s) such that the groups Gpi and Gpj are not iso-

morphic. Let H denote an abstract group isomorphic to the group G. By virtue of (1) the group H possesses an exact Z-representation F of degree n of the form

P : g -+ diag [A_4 (g)(t), Ao (g)(b o) . . . . . A~ (g)(b~)] (g C H),

where Ai is an irreducible Z-representation of the group H, ImA i = Gpi (i =--I, 0, 1,...,s).

Applying Lemma 7, we obtain that there exists an exact Z-representation ~' of degree n of the group H of the form

P' : g - ~ I" (g) = diag [F 1 (g), F 2 (g)],

where r i ( g ) is obtained from r(g) by discarding Ai(g) and Aj(g) in one block, and

P2 : g-~P2 (g )= (A~ (0g)Aj B(g) t(g)j i s a nondecomposable Z - r e p r e s e n t a t i o n of the group tt (A[ and Aj a re i r r e d u c i b l e Z - r e p r e s e n - t a t i o n s of the group H that are Q-equivalent, respectively, to the representations A i and Aj).

Let ImF' = G. We show that the groups G and G are not conjugate in GL(n, Z). We assume the opposite. Then a matrix CEGL(n, Z) exists such that

c - i t ' (g) C = I" (m (g)) (g C H), (2) v

where q0 is a certain automorphism of the group H. Evidently A i is an irreducible Zp-repre- sentation of the group H (i = --I, 0, I .... ,s). Hence, _r g-+r(q0(g))(gCH ) is a completely

�9 ! . . . .

reduclble Zp-representatlon of the group H. Consequently, we obtaln a contradlctlon from (2) and the valldity of the Krull--Schmidt theorem for the Zp-representation of the group H (see [11]). Therefore, the groups G and G are not conjugate in GL(n, Z).

2. Let all the irreducible components of the group G in (I) be pairwise isomorphic. Then according to Lemma 4, n = br( p -- 1)p r (I < b r < p, r ~> 0) and G = G(br ). Let N denote

P an abstract group isomorphic to the group Gpr. Then H = NI x ... x Nbr (N i ~ N; i = 1,...,b r)

is a group isomorphic to G. Let us consider the subgroup Nlx N2 of the group H. Let A be an exact Z-representation of the group N with ImA = Gpr. Then A is an irreducible Z-repre- sentation of the group N. Evidently

) I ' i : g i - + \ 0 A(gi ) , ' g2--~ ( 0

(gi C Ni, i = !, 2, and E is the unit matrix) is an exact Z-representation of the group Nl • N2 with Im•1 = diag [Gpr, Gpr]. The representations AI:gI-+E, g2-+A(g2); A2:g1-+ A(g~), g~-+E

of the group NI x N2 are irreducible and nonequivalent over Q. Then according to Lemma 7 the group NIx N2 possesses a nondecomposable exact Z-representation F 2of the form

, 0 A; (g)/

where A i is a certain Z-representation of the group Nl • N2 that is Q-equivalent to A i (i = I, 2). We hence obtain by the same reasoning as in case I that a Q-Sylow p-subgroup of the group G exists in GL(n, Z) such that the groups G and G are not conjugate in GL(n, Z).

The theorem is proved.

Remark I. It is shown in []6] that if K is a domain of principal ideals of character- istic p > 0, then any Sylow p-subgroup of the group GL(n, K) is conjugate to the group UT(n, K).

791

2. RELATION BETWEEN LINEAR GROUPS AND MODULI OVER GROUP RINGS

Let K Be a domain of principal ideals, G a finite group, and KG a group ring of th~ ring G over K. Henceforth we shall understand KG-moduli to be left KG-modu!i , which are free K- moduli of finite rank, Let M denote a KG-modulus of rank n in which an exact matrix K-repre- sentation F of degree n of the group G is realized. We shall say that the KG-modulus M cor- responds to the subgroup Im F ---- P (0] r (3 L(a, K)

Conversely, a matrix K-representation I' of the group G corresponds to each KG-modulus with a selected K-basis. We shall say the subgroup 7(G) of the group GL(n, K) corresponds to the KG-modulus M (with a fixed K-basis).

For an arbitrary KG-modulus M and an automorphism ~ of the group G we determine a KG- modulus M r' by considering that ,~4 ~ =M as the K-modulus and the operators g E G act in M Y according to the rule: ~r (rn~M) . The KG-moduli M! and M2 are called conjugate if there exists an automorphism ~ of the group G such that the KG-moduli A4~' and M2 are iso-

morph ic.

LEMMA 8. Let G i be a finite subgroup of the group GL(n, K) isomorphic to the group G (k = I, 2), and let Mi and M2 be KG-moduli corresponding to these subgroups. The groups G1 and G2 are conjugate in GL(n, K) if and only if the KG-moduli M 1 and M2 are conjugate.

Proof. Let r i be a matrix K-representation of the group G being realized in the modulus M i with respect to a K-basis such that ImF i = G i (i = I, 2). Let M~=A4~ , where %~EAutO and T is an isomorphism of the KG-moduli Mi and M2. Now ~ is defined in the selected K- bases of the moduli Mi and Mz by the matrix C C GL(n, K) such that CF~(9(g))=F~(g)C(~EO) Then C ImFi C-i = ImF2, i.e.,CGiC -i = G2. Conversely, let C~IC -~=O~(CEOL(I~,K)) . We set ~(.~) = pf-l(C.-IF~(g)C)(@6G). Evidently cp is an automorphism of the group G and the matrix C defines the isomorphism T of the KG-moduli A4~ and M2.

The lemma is proved.

It is easy to see that each automorphism 9 of the group G can be continued to the auto- morphism of the ring KG. Let the KG-modulus M be a left ideal of the ring KG. Then q~ (M) is also a left ideal of the ring [<G iq~ ~ Au~ KO)

LEY, MA 9. If a KG-modulus M is a left ideal of the ring KG, then the KG-moduli 9(M) and

]~i ~-I are isomorphic ,q0 ~ AutO)

Proof. Evidently g~c ~ ' ., the operator g acts in

,[ ~,~.) exactly as it acts in A4 ~-~.

The lemma is proved.

Let K be a field of characteristic zero. Then KG is the direct sum of minimal two-sided ideals of the ring KG that go over into each other under the effect of automorphisms of the group G. An irreducible K-representation of the group G which is realized in a minimal left ideal of the ring KG contained in V corresponds to each minimal two-sided ideal V of the ring KG. Let W n be a set of all minimal two-sided ideals of the ring KG to which exact irreducible degrees n of a K-representation of the group G correspond. Each automorphism 9 of the group G induces a substitution in the set W n.

Let r n be the nun~ber of irreducible pairwise nonconjugate subgroups of the group GL(n, K) isomorphic to the group G. On the basis of Lemmas 8 and 9 it is not difficult to prove the the following two assertions:

I) The number r n equals the number of orbits into which the set W n decays under the effect of the group AutG,

2) the number r n equals the number of pairwise nonconjugate K-characters of exact ir- reducible K-representations of degree n of the group G (the K-characters Xi and X2 of the group G are called conjugate if Z~:- Z~ for a certain qo~AutO)

3. REDUCTION OF THE PROBLEM OF DESCRIBING THE MINImaL IRREDUCIBLE SOLVABLE SUBGROUPS

OF THE GROUP GL(p, Z) TO A DESCRIPTION OF THE REDUCED SYSTEM OF POLYNOMIALS

Let us present the Suprunenko results [7] on the minimal irreducible solvable subgroups of the group GL(p, Q) (p is an odd prime) in a form needed for later.

792

Let Z 2 be a field of two elements, F = Za(e) (s is a primitive root of degree p of ]), [(x) = x m@ am_:xm-~-... @ a0 (a~Z~) is a polynomial irreducible over Z2, Whose root is e.

As is known, the number m equals the index to which 2modp belongs. Let G m denote a subgroup

of the group GL(2, F) generated by the matrixes

a= b---- ] '

Evidently Hm = UT(2, F) is a normal subgroup of the group Gm of index p. It is easy to see that Hm is an elementary Abelian 2-group of order 2 m. Let ~ denote an automorphism of the

group H m induced by an internal automorphism g--~b-lg.b (gCGm) of the group Gm. The elements a, oc~,...,om-z~ form a basis H m considered as a linear space over Z2. In this basis the Frobenius form of the polynomial f(x) corresponds to the automorphism o. Let F' be a sub- ring of the ring of all endomorphisms of the group H m generated by the element o. It is easy to see that F' ~- F. Each element h E Hm is represented uniquely in the form h=~c (~6F') ,

where c is a nonunique element of the group H m.

LEMYN ~0. To the accuracy of an inner automorphism generated by an element from Hm, each auto~orphism q~ of the group G m can be represented in the form q~(~a)----%~(a)(~6F'), ~(b)----b ~,

t~ ~ 2 ~ (mod p) (0 ~ ! < m)

Proof. Let

(; ~ Let us set

~=(1--s~) -:a, h=(10 ~ t . l /

Then h - % ( b ) , ~ = b k . L e t ~ = ~0 + ~1~ § ... § m-I (~iEZ2). Since ~ = ~ on Hm, then

(Za) = (6~ + ~l~k + ... + ~ _ ~ k ~ - , ) ~ (a) = ~'~ (a),

where X' i s an e l e m e n t of t h e f i e l d F ' o b t a i n e d f r o m X by t h e s u b s t i t u t i o n ~ + o k . I f Z = 0, t h e n X' = 0. I n p a r t i c u l a r f ( o ) ' = f ( o k ) = 0. T h i s means t h a t t h e c o r r e s p o n d e n c e X + ~ ' - + ~ ' (%EF') i s a n a u t o m o r p h i s m of t h e f i e l d F ' . T h e r e f o r e k 5 2 t (mod p) f o r c e r t a i n t (0 < t < m). Then X' = xk.

The lemma is proved.

LEMMA I] [7]. To the accuracy of conjugacy the group GL(p, Q) contains just one minimal irreducible solvable subgroup. This subgroup is isomorphic to the group G m.

Later to conjugacy accuracy we shall study subgroups of the group GL(p, Z) isomorphic to the group Gm. We first write one such subgroup down. We define a Z-linear character X of the group H m as follows:

% ( a ) = - - l , % ( e ~ ( a ) ) = l ( i = 1,2 . . . . . m - - I ) . (3)

It is easy to see that if h=~z6H m (~6F') , then x(h) = --I if and only if ~ = X0 + ~o + ~ + ~m-l~ rn-1 and %0 = ~ (%iEZ2)

Let us set

I 2 e = ~ ~ - - ~ - ~(h) h. ( 4 )

hEtIm

E v i d e n t l y e i s t h e m i n i m a l i d e m p o t e n t of t h e a l g e b r a s QHm and QGm. I t i s c l e a r t h a t he = 7~(h)e (hE Hm). We s e t

L = ZGme. (5)

I t i s e a s y to s ee t h a t e , b e , . . . , b P - l e i s a Z - b a s i s of t h e ZGm m o d u l u s of L. Then Z - r e p r e - s e n t a t i o n F of t h e g r o u p G m wh ich i s r e a l i z e d i n a modu le L w i t h r e s p e c t to t h i s b a s i s i s an e x a c t i r r e d u c i b l e Z - r e p r e s e n t a t i o n of t h e g r o u p Gm. T h e r e f o r e , t h e g r o u p ImP i s i s o m o r p h i c to t h e g r o u p Gin, and I m f ' c - G L ( p , Z ) . We f i n d t h e m a t r i c e s r(a) and F ( b ) . To do t h i s we g i v e

793

a function S in the following manner in the set of nonnegative integers. Let rk(x) be the residue upon dividing the polynomial x k by f(x) over the field Z~. We set S(k) -- rk(0). Evi- dently, • = _] if and only if S(k) = I (k = 0, ~,...). Since

able = b~';~ (c; ~ (a)) e = ( - - 1)s(~)J~e,

then

I" (a) -- diag [ ( - - 1) s(~ t - - 1) s(~ . . . . . ( - - 1)st~ <6)

LEI~4A 12. Le t G be an a r b i t r a r y subgroup of the group GL(p, Z) t h a t i s i somorph ic to the group G m. Then ZGm is the module M c o r r e s p o n d i n g to the group G, and is c o n j u g a t e to a c e r - t a i n ZG m submodule of the module L wi th r e s p e c t to Aut Gm.

Proof____. By v i r t u e of L e ~ a s 8 and t l , the QGm modules QM and QL a re c o n j u g a t e , i . e . , O M . ~ ( O L ) ~ (r~6 A ; ' i 0 ) . F u r t h e r m o r e , b e c a u s e of Lemma 9, (QL)r T h e r e f o r e H ~ : Q M ~ ~p-~(OL) . We set ]~{I := ~FT(M) . Then MI is a ZGm-submodule in QL, Since ~(M)=~-~(A4~)=M~ , then the modules M and M1 are conjugate. Since M~ is a finitely generated ZGm submodule in QL, then there is a ~EZ (==/=0) such that M~=o~M~_L . Evidently M~ and M2 are isomorphic

ZGm- modu I e s.

The lemma is proved.

COROLLARY 1. The set of all minimal irreducible pairwise-nonconjugate solvable sub- groups of the group GL(p, Z) agrees with the set of groups corresponding to all nonzero pair-

wise-nonconjugate ZGm-submodules of the module L.

LEMMA 13. Let M? and M2 be conjugate ZG m submodules in L, i.e.,M~_--~Ad=(~EAutG) �9 If M~ -: oqL -5 ... -4- O~sL (~z~6ZO~n), then z%4~ ---- l((p-I (~)L @ -.-~-~ (~s)L) , where X is a certain rational

number not zero.

Proof. Because of Lemma 9~:~-~(M~)~.44~4e The isomorphism ~ can be continued to

the isomorphism

Then

where e' = q~-~ (e) q~-~ (~) Z G , j ' (e') �9 au t omor phi sm

"d : Q ~ - I (M 0 = ~ - l (QL) ~ QM 2 = QL.

-c' (q~-I (QL)) --=- "~' (Qcp-~ (L)) = "~' (QG~.e') = QG,~z' (e') := QOe,

Since T' (~-~ (o; e)) = ~p--~(~)T' (e') (cr E QGr,~) , then ~ (~p-l (MI)) = ~F,--i (~i) ZG~ T~ (e') J-... + Let T'(e')=Be (~CQGm) . Then ~e:---c'f~o----e"~'(o'~~ ,~j . It is easy to see that for any

% of the group Gm

% (e) = ----

h6H m

and ~-i(e)----e (if %qct==7. ) or ~p-t(e) and e are orthogonal idempotents. It hence follows

that ~e=e'~e----~.e (ZCQ, %50). This means that #~2=~c(~p~(M1))=~.(~p-1(o~!)LnU...-iqD-l('z~)L).

The lerama is proved.

Remark 2. Under the conditions of Lerm-na 12, certain polynomials of b: ~i = ~i(b) can be selected as the generators ~i of the module Mz. There then results from Lemma 10 that

M~ ~ ~l (bk) L + ... -t- cz, (b k) L,

where k = 2 t for a certain t (0 ~< t < m).

Let Z2k be a ring of classes of residues modulo 2 k, and K = Z2kG m a group ring of the group G m over Z2k. Evidently R = 2K + (~- I)K is a radical of the ring K. Let us set V = L/2kL. It is easy to see that RV ~ 2V. Then V is a K-module and every K-submodule in V can be considered a ZGm-module. It is easy.to see that 2iv (0 -<. i < k) is an Abelian group of type (2 k-i . 2k-i). In particular, 2zV contains all elements of the group V whose orders equal'2k'i~ " " '

794

Let us first study the structure of the ZG m submoduli in V for k = ~. In this case the group H m acts trivially in V.

LEbIMA 14. A ZGm-module ~ = L/iL can be considered as a regular ZiB-module (B = <b>) in which Hm acts trivially. Any ZGm-submodule N in L is cyclic and is generated by an element g(b): N = g(b)L, where g(x) is a divisor of the polynomial • I modulo 2. For each divisor g(x) the module N = g(b)~ is a ZGm-submodule in L.

Proof. ~ is a trivial Hm-module with basis e, be,...,bP-~e (e = e + 2L). This means that L is a regular ZaB-module. Since (IB], 2) = I, then ~ is a completely reducible module. The factor ring Z2 [x]/<xP -- I> can be considered a ZGm-module in which H m acts tri_vially, and the element b as a multiplication by x. It is easy to see that Z2[x]/<xP -- I> m L and under this isomorphism the ideals in the ring Z~[x]/<xP -- I> correspond to ZGm-submodules in ~ and conversely. All the rest follows from the structure of ideals of the ring Zi[x]/<xP -- I>.

The iemma is proved.

On the ring Z2 Ix] we define a Zi-linear function T by setting T(x n) = S(r)x r, where S is the function defined above, and r is the remainder from dividing n by p (r >~ 0).

LEMMA ~5. Let g(x) EZ[x] Then (a-- l) g(b)e----ime ,where (o{ZB (B= (b)} and ,~ = T(g(b)) (the bar denotes reduction modulo 2).

Proof. From (6) it follows that (I --~g)bre = 2S(r)bre (r = 0, ~, 2 .... ). Then

1 (! - - a ) b ' e = T(b%e. 2

The lemma is proved.

Let us introduce a Zi-linear function D in the set of polynomials Zi[x] by setting D • (g(x)) = nCD{g(x), T(g(x)), T(xg(x)),...,T(xP-lg(x))} (g(x) C Zi[x]). We call the system of nonzero polynomials g0(x) .... ,gr(x) of the ring Zi[x] reduced if the following conditions are satisfied: I) g0(x) is a nontrivial divisor of the polynomial xP -- I; 2) gi+l(x) is a divisor of D(gi(x)) (i = 0, I .... ~r- I).

Now, let us examine the structure of the submodules in V = L/ikv for k > ].

LEMMA 16. Let N be a K-submodule in V which contains none of the modules 2iv for 0 < i < k. Then there exists a reduced system of polynomials g0(x),...,gk-1(x) and elements ~0,.-.,~k-i of the ring ZB such that

N = (%g + 2 0 ) y q- . . . Jr 2 k - l ~ % _ i V ,

~i = g i (b) (i = 0, 1 . . . . . k " 1).

Proof. We set N~=N N 2*V (i----0 1 ..... k--l) Then N i is a K-submodule in N = No, Ni+z =

Ni ~ 2i+IV and NI+I ~ 2~+IV . We show that in the series

N = N o ~ N1 ~ . . . ~ N ~ - I ~ N k = 0 ( 7 )

all the factors Ni/Ni+ l (i = 0, I .... ,k- I) are isomorphic to nontrivial K-submodu!es in L = L/iL. Indeed, for 0 ~ i < k

NJNi+l = NJNI I] 2~+1V - - (N~ Jr 2~+1V)/2 i+ lg _~ 2'V72 ~+l g _~ V/2V ~-- L/2L : L. (8)

We show t h a t N i / N i + z ~ 0 and N i / N i + 1 ~ L ( i = 0 , 1 , . . . , k - 1 ) . L e t N i / N i + ~ = O. Then N

2 tV~ 2~+lv. I f ~oCN a n d m~2e+l V , t h e n m~2~V . T h e r e e x i s t s a l e a s t t ~< k such t h a t 2%~C2~V.

l~nen 2%)E2e+lV and 2t -%)E2~V+_2~-IV~2eV , w h i c h c o n t r a d i c t s t h e s e l e c t i o n o f t . T h e r e f o r e , Ni/Ni+1 ~ 0. Let Ni/Ni+z ~ L. Then

(NvO r- 2i+1V)/2~+"V = 2tV/2~+lV,

i.e.~ N i + 21+iV = 2ZV. The module V is a cyclic Z~kB-module and is generated by the element = e + ekL. Let 2i~ = 0~i + 2i+iw2, where ~ N,~Af, 0~.a{V . Then 2k-~e = 2k-i-~wz ~ N and

2k-Iv ~ N, which also contradicts the conditions of the lemma. Therefore, Ni/Ni+ 1 ~ ~. Thus, all the factors of the series (7) are isomorphic to the nontrivial ZGm-submodules in

= L/iL. By virtue of Lemma 14, for each i (i = 0, 1,...,p -- I) there exists a nontrivial divisor gi(x) of the polynomial ,~--[ (gi(x)~%~l:~i) such that N i + 2i+Zv = 2igi(b)V + 2i+!v. From (8) there results the existence of a mi~9 ~ such that

795

N~ = 2 % y + ,'V~+1. (10)

SinceRN~c2N~+t, , then gi(b) = Igi+l(b), and because of Lemma 15, T(bag~(b)) =X~g~4_1(b) (tz----O, i .... : ~, ).~EZ~B). Hence, taking into account that the degree of 'i'(g(x)) (E (x) E Z2 [x]) is less than p, there results that gi+1(x) divides D(gi(x)), i.e., g0(x),...,gk_l(x) is a reduced sys- tem of polynomials. There results from (10) that N = No = {#0V + 2~IV + ... + 2k-~,~k-iV.

The lemma is proved.

THEOREM 2. For every nonzero and nonisomorphic to L, ZG m submoduleMof the module L there exist a natural number k, a reduced_system of polynomials g0)x) ..... gk_i(x), and such elements ~0,...,~k-I of the ring ZB that w i = gi(b) (i = 0, 1,...,k- I) and

M ~- %L + 2ohL + ... + 2t'-4o)k-lL + 2'~L.

Proof. Let d(M) denote the greatest common divisor of the coefficients of Z-forming modules M in their expansion in a Z-basis of the module L. Then M~d(M)L . Let Ml = [I/ d(M)]M. Then ]V.t--M and d(M1) = I. Let e i = bieb -i (i = 0, 1,...,p -- I). Then e 0 = e, el,...,ep_ l are pairwise orthogonal idempotents in ZH m. It is easy to see that eibke = ej for k _ j (modp) and ejbke = 0 if k } j (modp). Since 2me~ZHm , then 2%~Mt~ z}i �9 Let

~--I

e) = ~ %ib~e~ M t . Then 2%,0~ = 2'~,~b~e~Mt �9 T h e r e f o r e 2~X~e~,4/l~ (] = 0 1 . . . . . p - - l ) Then 2md(M~) x

e = 2 ~ e ~ M , i.e., 2mL~M, There exists a least natural k such that 2 ~ L ~ M t . We set V = L/2kL and N = Mz/2kL. Then N is a submodule in V that satisfies the conditions of Lemma 16.

The proof of the theorem hence follows at once.

LEMMA 17. Let M? and M2 be ZGm-submodules in L and let rP be an automorphism of the group G m such that M~_--~ .M= . Then the modules Mz and M2 have identical reduced systems of

polynomial s.

Proof. If g0(x),...,gk-z(x) is a reduced system of polynomials of the ZGm-submodule M in L, then the elements g0(b),...,gk-z(b) generate factors of the series (7) of the module N = M/2kL. It is easy to see that if g(x)6Z~[x] , then g(b)~ = g(b2)[ (~ = L/2L). Therefore, the substitution b + b 2 conserves all the factors of the series (7). Theorem 2 and Remark

2 complete the proof of the lemma.

4. FOUNDATION OF THE ALGORITHM TO CONSTRUCT REDUCED SYSTEMS OF POLYNOMIALS

Let f(x) be a nontrivial divisor of the polynomial xP -- I over the field Z2. Let us define S in the set of nonnegatiVe integers by the following method: S(n) = rn(O), where rn(x) is the remainder from dividing the polynomial x n by f(x) over Z2. We denote by T a Z2-1inear function on Z2[x] such that T(0) = 0, T(x n) = S(k)x k, where k is the remainder from dividing

n by p (k -> 0). If E(x)EZ~[x] , and g(x) ~ 0, then we set

D' (~) = LCD { T (g (x)), '/' (x~ (x)) . . . . . tr ( x ~ (x))}, D(g) = LCD {g(x), D' (g(x))}.

When f(x) is an irreducible divisor of ~p(X) over Z2, then S, T, D are functions defined

in the previous section. If g(x) = ~0 + ~! x + -.. + er x' (r <~ p), then we set

p--r g* (x) = o~o,'g 4- ~ Z - t ~ ... + c~,.~: ..

LEMMA 18. Let b(x) = D'(%p(X)), then h*(x) = If(x) + 1][(xP -- 1)/f(x).

Proof. Let m be the degree of the polynomial f(x) and

l ~ - F ' o o ' F ~o,,x '~ .... + , o . . . . t'

1 " ~ . o + # t , 1 x + + . . . . . z . . . 31 ,tr~_l X

. . . . . . . . . . . . . . . } (mo , J f (x)) X p - I - r I_ m--I

~n_l~,co -F" ~O_1,1 X-, - - . + ~ _ . l , m . _ l x i p . . . . m--1

x = ~'p,o + ~o,,-~ + --- + ~'. . . . . ~x J |

(11)

796

e r

of (11) b e d e n o t e d by h 0 ( x ) , h i ( x ) , . . o , h p ( x ) , r e s p e c t i v e l y . E v i d e n t l y h o ( x ) = ~o ,0 = 1; hp(x) = ~p,0 = I. It is easy to see that the following recursion formula is true

hi+: (x) = x ~ (x) - - 13~,,._:f (x),

whose p - t u p l e a p p l i c a t i o n y i e l d s

~ (x) = x ~ -- ~ (x) ( .~o,m_S -~ ~ + ~p_~,,._~x + ~_: ,~_~) .

From (12) there results that ~i+i.0----~.m-: (i = 0, !,...,p- I). Therefore

~, (x) = x ~ - / <x) (~o,o x~ + ~,o x~-~ + ... + f~_:,ox) + (~o,o x~ - - ~,o) f (x)

or

Then T(x')----~r.0 xr (O~-~r~p) and h(x) ~0.0@~,0x~ - ~_:,Qx g- . Let the right sides

I = : s (x)h" (x) + (x ~ - - t) f (x) .

The lemma is proved.

Let g(X)=yo--{[email protected]_:% p-~ (?~EZ~) . Then

(12)

p--I

T (xig (x)) -~- Z Yi-i[3~.~ (mocl (x ~ - - 1)) ( 13 ) f~=O

( , ~3 = 0, 1,...,p- I), where operations over the indices are performed in Zp = (0, 1,.o.,p-- I}.

Let ~ be an arbitrary root of the polynomial xP -- I in an algebraic closure of the field Z2 �9 Then

p--I . . p - - i p - - 1 / p - - i

i = 0 /=0 f=0 ", ~=0

From (13) there results that

p - - [

g (x) i~ ~ (~-:x) ~-- ~ T (~f-ig (~)) x i (mod (xo _ 1)). (14) f=0

LEMMA 19. Let f(x) be an irreducible divisor of xP -- I over Z2 and ~ a root of f(x). The polynomial f(x) divides D(g(x)) if and only if f(x) divides g(x) and g(~x) o

Proof. Let f(x) divide D(g(x)). Then f(x) divides g(x) and D'(g(x)), i.e., f(x) divides all polynomials T(xig(x)) (i = 0, 1,...,p- I). Therefore, T(~ig(~)) = 0 (i = 0, ]~...,p- I). Then from (14) and Lemma 18 there results the equality

xP--1 .

g ~x) ~ q (x) + 1) = q (x) (xo - - 1),

{q(x)EZ2(~)[x]) , from which the divisibility of g(~x) by f(x) follows.

Conversely, let f(x) divide g(x) and let f(x) be divided by f(~-lx). Then in conformity with Lemma 18, the left side in (14) is congruent to zero modxP -- I, Which is possible only if all the coefficients on the right side are zero. Therefore, ~ is a root of each of the polynomials T(xJg(x)) (j = O, 1,...,p- I). Since ~ is an irreducible root over Zz of the polynomial f(x), then f(x) divides all T(xJg(x)), i.e., divides D'(g(x)) and D(g(x)).

The lem~na is proved.

We call the number k the length of the system g0,...,gk-:. We let I k denote a set of systems of length k. Evidently I~={g(x) EZz[x]lg(x ) divides xP -- I and g(x) = xP -- I}, Iili = 2 s+: -- 2, where s is the number of irreducible polynomials over Z2 whose product over Z2 is a polynomial of division of the circle ~p(X). It is easy to see that if f(x) = Op(X), then the set I: exhausts all the reduced systems of polynomials. Let s > I and let f(x) be an irre- ducible divisor over Z2 of Op(X) and let ~ be a root of the polynomial f(x). From Lemma 19 there results the existence of reduced systems of length k = 2 of the form

(x - - 1) f (x) g (x), x - - 1, ( 15 )

797

where g(x) is an arbitrary divisor of ~p(X) not divisible by f(x), and such that (x -- l)f(x) • g(x) = xP- I. The number of such systems is 2 s-~ - I [the number of divisors g(x) of the polynomial ~p(X) not divisible by f(x)]. From this same lemma there follo~s the existence of reduced systems of the form

q)~ (x), l(x), (~6)

where t(x) is an irreducible divisor of ~p(X) such that s -~ is not a root of t(x)(t(~ -~) ~ 0). The number of such systems is s - I. We denote the set of systems of the form (15) and (16) by I~. We set f=I~U f's �9 Evidently IIl -- 5.2 s-~ + s- 4.

5. APPLICATION OF THE ALGORITHM TO OBTAIN LOWER BOUNDS OF THE NUMBER OF CERTAIN

CLASSES OF IRREDUCIBLE FINITE SUBGROUPS IN GL(p, Z)

Let ~p(X) = f~(x) ... fs(x) be the decomposition of the division polynomial to the circle ~p(X) into a product of irreducible polynomials fl,...,fs over the field Z2, whose degrees evidently equal m = (p -- 1)/s. This decomposition can be continued modulo any 2 k (k > l). Let ~)~(x)~f~(x)...f~(x)(mod4) be the decomposition of ~ (x) over the ring Z~ where fi(x) - p fi(x) (rood2) and the highest coefficients of the polynomials fi(x) are one (i = 1,...,s). Evidently the degree of the polynomial f;(x) also equals m. Let

7=0

be elements of the ring ZB (see the notation in Lemma 14) such that mi ~ f[(b) (rood4) (i = 1,...,s). It is easy to see that

~t...,~---__=cD~(b) I (rood4). (17) (b-- 1 ) ~ ... ~ - - 0 [

We call the element ~i corresponding to the polynomial fi(x).

LEMMA 20. For each system (u(v), v(x)} belonging to I�89 there exists a unique ZGm-sub- module in L to eonjugaey accuracy, whose reduced system of polynomials agrees with the given

system.

Proof. Let t and r be powers of the polynomials u(x) and v(x). Then I ~< r < t ~< p - I. We note that the polynomial v(x) is irreducible over Zz. We set u'(x) = (xP- 1)/u(x), v'(x) = u(x)/v(x). All the polynomials u, u', v, v' are products of certain of the polynomials f0(x) = x- I, fl(x),...,fs(x). We let ~, ~', n, D' denote the product of ~i(0 ~< i ~< s, ~0 = b -- I) such that they correspond to those polynomials fi(x) whose products are u, u', v, v', respec- tively [0 ~ i ~< s, ~0 corresponds to f0(x)]. From (17) there follows that

~ln.' = ~, o)'~ ~ 0 (mod 4). (18)

Let M denote a Z-submodule in L with the generators

o)e, bo)e, ..., b~- t - l o~e,

~,,_~_ (19) 2qe, 2brle . . . . ,20 qe,

4e, 4be . . . . ,4b~-~e

[see the notation (4) and (5)]. It is easy to see that the system of elements from L that have been obtained from (]9) by discarding the factors 2 and 4 in the second and third rows is a Z-basis in L. It hence follows that the system (19) is a Z-basis in M and 4L ~ M. We show that M is a ZG m submodule in L. To do this it is sufficient to show that bM ~ M and (a--I)M~M. The condition bM ~ M results from (18). From Lemma ]5 the existence follows of @iEZB, such that (a- ])biwe = 2@ie and e i _= T(biu(b)) (rood2). Since v(x) divides T(xlu(x)) (the condition of reducibility of the system u, v), then there exist elements ~i% ZB such that T(biu(b)) -n ~ni (mod2). Therefore,(a--i)b~o~ee~2qqie(mod4L) , i.e.,(a--1)bioe6M Since (a--1)L~2L , then 2(a-- l)bi~e 0 (mod4L) and 2~a-- 1)i~eCM Therefore (a--I)M~-M and M is a ZGm submodule in L. Since 4L~M and M!!~L , then the length k of the reduced system of polynomials for M (see Theorem 2) equals either ] or 2. The case k = I means that 2eCM. This is possible only when2==~i~@2N%~-4L3(%i,%~,s Hence there results from (18) t h a t 2o'------- 2o~'n.)~ (rood4) or ~ ' ~ ' ~ t ~ 2 ( m o d 2 ) Then v ' ( x ) ~ - - v ' ( x ) u ( x ) s 1)) (s

798

which is impossible since u(x) is divided by one of the polynomials f0, fl,.-.,fs- Thus, the length of the reduces system of polynomials for the module M equals 2. Evidently this system agrees with [u(x), v(x)}. The existence of the ZGm-submodule required by the lemma is proved. Let us prove its uniqueness. Evidently the module M is generated by any system obtained from (19) by replacing ~ by NIEZB (N1-------N(mod2)) . Let M' denote a ZGm-submodule in L generated by a system obtained from (19) by replacement of ~ by ~i----~0-~2g(b) (g (x) 6 Z2 [x]) Let g(x) and v(x) be mutually prime modulo 2. Then u'(x)g(x) is also mutually prime with v(x). There exist %1(x), %, (x) C Z[x] such that k1(x)u'(x)g(x) + %2(x)v(x) = I. Since u'(b)~--~',v(b)~B(mod2), t h e n ~ (b) ~ ' ~ + 2~ (O)~ t-- Z~ (b)o)'o~ ~- 2~ (b) ~ 'g (b) ~- 2~ (b) ~ ~ 2 (nod 4),. S i n c e M' i s a ZGm-submodu! e in L that contain 4L, then 2e6M' . Therefore, the length of the reduced system of polynomials for M' equals unity, and this system cannot agree with {u(x), v(x)}. If g(x) is not mutually prime to v(x), then from the irreducibility of v(x) there follows that g(x) is divided by v(x). In this case, M' = M as is easily seen.

The lemma is proved.

THEOREM 3. For a number d of piecewise nonconjugate minimal irreducible solvable sub-

p-I ~_~ p - - I - -3 is valid where m groups of the group GL(p, Z) the following bound d~5X 2 ~ fft

is an exponent which belongs to 2 (nod p). All these subgroups are isomorphic to the group G m. The bound is reached in the cases m = p- I and m = (p- I)/2.

Proof. For each ~6I we let L~ denote a ZGm-submodule in L, whose reduced systemagrees with k. If k ---- {u, v} ~l_~ , then the Z-basis LX agrees with (19). If %~1~ and ~ = u(x), then we define the Z-basis L~ by the system

o~e, b ~ e . . . . . b p - t - ~ (oe, 2e, 2be . . . . 2b ~-~o~e. (20)

From Lemma 17 there results that the modules L~(~I) are pairwise nonconjugate, and each is not conjugate with L.

Let F~(k~l) be the Z-representation of the group G m realizable in the Z-basis (19) or (20) of the module L~. Then the groups ~(Ora),F~(O~) ('~61) are pairwise nonconjugate in GL(p~ Z) (see Corollary 1). Tbe number of these groups equals [I[+ I = 5"2 s-~ + s -- 3 [s = (p -- ~)/m, where m is the exponent to which 2 (mod p) belongs]. If m = p -- I, then I2 = {~} and I = Iz, [Izi = 2. The corresponding ZGm-modules of Lk have the form

L . _ ~ = (b - - 1) L n t- 2L,

The number d equals 3.

Let m = (p- I)/2 and fl(x)f2(x) = ~p(X)

L%~.) = (I)p (b) L + 2L. (21)

in the ring Ze[x]. Let the polynomial f(x) used to define the group G m agree with fl(x). Then 11={x--1, fi(x),[2(x),(x--l)[,(x),(x--l)f2(x)cD~(x)] , the set I~ consists of two systems W----{(x--l)[1(x),x--1 } and {(D~(x), It(x)} if p ~ I (nod4), W and {(Dp(x),/~(x)} if p - -I (nod4). There are no other systems. The corresponding modules of L% have the form

L x _ l - - - - ( b - - 1 ) L + 2 L , Lf ,= [~(b) L + 2 L ,

Lr, = [2 (b) L -~ 2L, L~o ---- q~p (b) L + 2L,

L(x_ i ) f i = (b - - 1)fj(b) L + 2L (] = I, 2), (22)

L~-l)r,~-I ---- (b - - 1)[~ (b) L + 2 (b - - 1)L + 4L,

L~p.f~ ----- <Dp (b) L + 2[i (b) L + 4L

(i = I or i = 2). The number d equals 9.

The theorem is proved.

Remark 3. Let T% be the transition matrix from the basis e, be,...,bP-le of the module L to the basis (19) [or (20)] of the module L%. Then F~(g)=T~IF(g)Tz(g~Gm) . In particular

r~ (On) = (r~. (a), r~ (b)>. (23)

THEOREM 4. The g roup GL(z, Z) c o n t a i n s n o t l e s s t h a n 3s + 1 f o r s > ] and no l e s s t h a n 3 for s = I pairwise-nonconjugate maximal irreducible finite subgroups in GL(p, Z) [s = (p -- 1)/m, m is the exponent to which 2 (modp) belongs].

799

Proof. Every maximal irreducible finite subgroup ;in GL(p, Z) is a group of invariants of a certain positive-definite integer quadratic form (see [2]). Two integer quadratic forms with matrices A and B of order p are called similar if there exists a matrix CEOL(p, Z} such that C'AC = %B, where I is a rational number (I = 0). Utilizing the absolute irre- ducibility of finite irreducible subgroups of the group GL(p, Z), it is easy to show that two maximal irreducible finite subgroups from GL(p, Z) are conjugate in GL(p, Z) if and only if their corresponding integer quadratic forms are similar. There hence results that for the conjugacy of two maximal irreducible finite subgroups in GL(p, Z) it is necessary that the ratio of the determinants of matrices, the integer quadratic forms corresponding to these subgroups, be the p-th power of a rational number. The groups P(G m) and F~(O~ (%El) remain positive quadratic forms with matrices E and A~=TITx(~EI ) , respectively (see Remark 3).

Evidently detA~=(detTj ~ . Let ~EI and X run through the following values

x - I, h (x), h (x) h (x) . . . . . h (x) h (x) ... L (x),

( x - - 1) h (x), ( x - - 1) h (x) h (x) . . . . . (x - - 1) ft(x)... [~-, (x).

From (20) there follows that for these I the det T 1 takes such values

4,4m, 4mn,.. . ,4m~,4~+l, 4 2jn+l . . . . , 4 '~'(s-l)+l.

T Now let ~ take on the following values from I2:

= { % (x), v (x)}

[ v ( x ) i s an i r r e d u c i b l e d i v i s o r o f ~p(X) o v e r Z2]

X, = { (x - - t) [1 (x) ... f~ (x), x - - 1 } (i = 1, 2 ..... s - - l ) .

From (19) t h e r e r e s u l t s t h a t f o r s u c h X t h e d e t T x t a k e s on t h e f o l l o w i n g v a l u e s

detTxo = 4'~(~-~ l) = 4~ m-J, d e t T ~ = 4 '~+ 2

(i = t . . . . . s -- 1).

(24)

(25)

It is easy to see that none of the numbers of the sets (24) and (25) and none of the ratios of these numbers is a p-th power of a rational number. Therefore, no two quadratic forms with matrices E and T Z (% runs through the mentioned s + I + s- I + s = 3s values) are simi- lar. If m = p- I, then the forms with matrices E, Tx-1(detTx-1 = 4), T~(x)(detT%(x)----4 p-I) are also not similar. Therefore, the maximal irreducible subgroups of the group GL(p, Z), which are the groups of invariants of the quadratic forms considered, are not conjugate in GL(p,

Z).

The theorem is proved.

Remark 4. As results from [5], for p = 7 the bound in Theorem 4 is reached.

6. APPLICATION OF THE ALGORITHM TO DESCRIBE FINITE IRREDUCIBLE SOLVABLE

SUBGROUPS IN GL(p, Z)

Suprunenko [7] showed to conjugacy accuracy that GL(p, Q) contains a unique maximal fi- nite irreducible solvable subgroup. This subgroup is a semidirect product H >x BI of an ele- mentary Abelian 2-group H and a group B1 of order p(p- I). The group B l is isomorphic to a group with two generators b and c and the governing relationships b p = cP -l and cbc -l = b ~, where v is a primitive root modulo p. The center of the group /f>~BI is generated by the matrix E and is extracted by the direct factor.

Let G be a finite solvable irreducible subgroup in GL(p, Z), then G is isomorphic to the subgroup in H)k BI. We shall consider the center of the group G trivial. Evidently it is sufficient to describe only such groups G. All the rest are obtained by adding the matrix E to G as generator. The group G contains a subgroup isomorphic to the group Gm. Moreover, the group G possesses a normal subgroup H I which is a cyclic expansion of order p of the max- imal normal elementary 2-group in G. We call the subgroup HI the basis subgroup of the group G. It is easy to see that G is a semidirect product of HL and the group <c(P-1)/k> of order k, where k divides p- I. For the rest, it is sufficient to limit oneself to the considera- tion of just those groups G whose basis subgroups are isomorphic to G m or the group

800

G~_, :H,,-i >x (b), [ff~_i!=2 ~-~ , and in a certain basis of the group Hp_ i the endomorphism ~(o(h):b-~hb, h~H;-~) agrees with the Frobenius form of the polynomial %p(X) o If the basis subgroup of the group G is isomorphic to Gin, then

(k is any divisor of p -- i).

p-I ,(,~) ' -E

G ~- O,v-i = G , - I >., (c ), (26)

If the basis subgroup of the group G is isomorphic to Gm~ then

r)--I

G=~G~ ) =G~>,.(c k ), (27)

where k is a divisor of m. Indeed, the order k of the group <c(P-l)/k> should be a divisor of the order of the Galois group of the field GF(2 m) (see Lemma 10). If the exponent m to which 2 modulo p belongs equals p- I or (p- I)/2, then all the solvable irreducible finite subgroups in GL(p, Z) whose centers are trivial are exhausted to isomorphism accuracy by the groups (26) and (27).

LEMMA 21. Every irreducible Z-representation, of degree p, s of a group G is the con- tinuation of a certain irreducible Z-representation F of the basis subgroup Gz of the group G. If for a given representation F of the group GI its continuation A to the representation of the group G exists, then there are not more than two of these continuation, ~/nere the sec- ond continuation exists only for even [G:GII and is obtained from A by replacing the matrix A(d) by-5(d) (dG1 is the generating element of the quotient group G/GI).

Proof. The group G~ contains a minimal irreducible subgroup G m. Since F is an abso- lutely irreducible representation of the normal subgroup G l in G, then its every continuation on the group G is obtained from A by tensor multiplication by a one-dimensional representa-

tion X(Z(@)= I, gEOl, z(d)=~,~ 'a:~ : I) of the group G,

The lemma is proved.

Let X1 be a nontrivial Z-linear character of the group Hp-l

' Z ez - [Hp-1 [ Z~ (h) h, L (" = Z G - l e ~ .

h6H

LEtKMA 22. All pairwise nonconjugate subgroups in GL(p, Z), which are isomorphic to the group Gp-1, are in one-to-one correspondence with the orbits on which the set of nonzero ideals of the ring Z2B decomposes under the effect of the group Aut B x (B = <b>).

Proof. If G~GL(p,Z~ and G m Gp-l, then a certain ZGp-I submodule of the module L (1) corresponds to the group G. It is easy to show that every ZGp-l-submodule M in L (I) (M L (1)) is determined by a certain reduced system of polynomials constructed for the polynomial ~p(X). As already noted, the set of reduced systems for .6p(X) agrees with the set I~ of non~ trivial divisors of the polynomial Cp(X) over the field Z2. Hence, it can be considered that 2L(1)~M and the module M/2L (I) is isomorphic to the ideal of the ring Z2[x]/(xp--] ) gener- ated by a divisor of the polynomial xP -- I. It is easy to see that AutGp_ 1 induces all auto- morphisms of the group B. Hence, the proof of the lemma follows.

COROLLARY 2. If 2 is a semiprimitive root modulo p then GL(p, Z) contains evidently 5 nonconjugate subgroups isomorphic to the group Gp-1. These subgroups correspond to the fol- lowing ZGp-I modules

L <1~, (b -- I) L (D + 2L (b, (Dp (b) L (D + 2L (I~, ( 28 )

fl (b) L ~ + 2L (~, (b - - 1) h (b) L (l~ + 2L (D.

If F (1) is a Z-representation of the group GO- I realized in the basis el, bel, ..~.bP-lei of ( 1 ) --i ~ ( I ) �9 ( [ , " the module L (l) , then F(1)(Gp_1), F~ (G~_I):T;~ F (G~_I)T~(~:x--I b, ~x), /1(x), (x--l)f1(x)) are all

the piecewise nonconjugate subgroups in GL(p, Z) isomorphic to the group.Gp-!. The character X~ of the group Hp-! can be given so that U(D(a)=diag[--],l ..... I,--I] (a is a nonunit element of the group Hp-~). Then

F[ l~ ((},_~) ---- (Pj]) (a), I~i ~' (b)>, (29 )

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According to Lemma 21, to describe finite solvable irreducible subgroups G of the group GL(p, Z) it is necessary to describe (to conjugaey accuracy) all the basis subgroups of the group G and to seek all Continuations of these subgroups. Theorem 3 and Corollary 2 [see (21)-(22)] yield a description of all nonconjugate basis subgroups in the cases when 2 is a primitive semiprimitive root modulo p.

Let C be a substitution matrix corresponding to the substitution

of a set of elements of the field Zp. It is easy to show tha~ generating elements of the subgroups Hm or Hp- l in the group G, of the form (26) or (27) can be selected so that the characters X and Xz Of these subgroups, which are used to define the representations F and F (1), will be invariant relative to the action c(P-Z)/k (see [8]). Then setting c(P-Z)/ke = e (or c(P-~)/ke~ = e~), we convert the modules L and L (I) into ZG-modules. This means that the representations F and F (~) of the groups G m and Gp-z allow of continuation to Z-repre- sentations of the groups (26) and (27). It is easy to see that the matrix C(P-I)/k will cor- respond to the element c(P-l)/k in these continuations. All the considered submodules in L and L (~) are converted into G-modules, with the exception of L (~) and L(.l ) . , zn the case

( x - ~ ) f ~ when k = p- I. It is easy to see that

Cr(1) ICD r ( l )

C) (1) (D oL(x- i l f~ =- L~.~)f2 =/= L(~-bf z.

Taking account of these remarks, the classification of the basis subgroups [see (23) and (28)] and Lemma 21, we obtain the following description of finite irreducible solvable

subgroups in GL(p, Z).

THEOREM 5 [17]. Let 2 be a primitive root modulo p. Then the group GL(p, Z) possesses just the following pairwise nonconjugate finite irreducible solvable subgroups

D--I

S~,o= (P(G~,_~),C~-), Sk,oX (--f),

Si~,~ = <r~. (0~_~), C~f ), S...,~ X ( - - E), b'--I

S'~.,~ = (L . (6~ , - I ) , __,%.k' ), 0--1

S~.,,,o = (1" (0~_I ) , - -C '~" ),

where k is a divisor of p -- I, k' is an even divisor of p- I, % C [x -- l, r (r C~=T~.IC'P~. The quantity of these subgroups equals 6< + 3~', where ~(T') is the number of all divisors of the number p - I [or (p - I)/2]. The groups Sp-LoX (--E), S~-~n~X(--E) are the maximal finite irreducible solvable subgroups in GL(p, Z) (see the notation in Remark 3).

THEOREM 6. Let 2 be a semiprimitive root modulo p. All finite solvable irreducible subgroups of the group GL(p, Z) are exhausted to conjugacy accuracy by the following sub-

groups p--I

U h = (F (G,~,), C -k-- ), b'j, X ( - - E), p--1

U~,:, - - (Fx(G~), C;~ ~ ), Uk.~.>< ( - - E ) , p--I p--!

u'~ ,~ = ;~'-,. ,. <G,~O,-- C~=) , U.~, = <~'~ ( C ~ ) , - - C T )

t (~; =:,i_!~-," /~ where LL and 12 are presented before formula (22); k is a divisor of p , I, and k' is an even divisor of p- I),

p--~

V,~ := (1 "':1; (O~_,), C ~ ~, V~ )/~ (-- ~7), p--'[

�9 - i ~ , , , ' , i i i, ~ : ,-, k ~T v : ~ . . ~-= ~l~ . .u~ ,_~p , t ,~ ) , ~' k,.,. ."-< ( - - E ) ,

802

p-I

G.,x : ( G ! t ( G - ~ ) , - - C~)' ),

'/~" = ( i ~') ( < b - P , - - C ~' )

(}~E{x ,~,~ . -- even- --' ( b , ~ s ) } , ~-- divisor p--l, ~.' odd divisor p-- t ) ;

p - !

(I;, (G-O, ~, ), Wk,~X ( - - '-,, p--I

W,~,>, : /~(n,. , C k"

where %~ { ' ~f!vr), (x -- l) [1(x)} , k is a divisor of (p- I)/2, k is even divisor of (p- I)/2. The quantity of these subgroups is 6T + 25T' + 11~", where T, T', T" denote the numbers of all divisors or p- !, (p- I)/2, (p- I)/4, respectively, The groups

Up_~ ~ X<--E>, X(--E), -~--,u.~'p{ ~),f~(x) i UP-2t , {(x-l)fl (~),~-I }

G-I X (-- S>, G-,,~-, X, (--S),

G-,,%(~, X (--E),

are the maximal finite irreducible solvable subgroups in GL(p, Z).

Let us note that V. M~ Glushkov approved the authors' investigations on utilizing an electronic computer in the theory of integral linear groups. The authors are deeply grate- ful to Yu~ V. Kapitonova and A. A. Letichevskii for a number of valuable comments.

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