Introduction to Combinatorial Topology-Reidemeister (1932)

8/11/2019 Introduction to Combinatorial Topology-Reidemeister (1932)

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arXiv:1402

.3906v1

[math.HO

]17Feb2014

Introduction to Combinatorial Topology

Kurt ReidemeisterTranslated by John Stillwell, with the assistance of Warren Dicks

February 18, 2014
http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1http://arxiv.org/abs/1402.3906v1


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Foreword

The first three chapters of this book deal with infinite groups; the last four with lineand surface complexes and especially with 2-dimensional manifolds. This choice ofmaterial may be justified from several standpoints.

In the first place I was concerned to workout the profound connections betweengroups and complexes. The close connection between these two fields has beenknown since the basic work of HENRIPOINCAR. If it has not been plainly evident inthe further development of combinatorial topology, then this is due to the problemswhere topology and group theory meet: it seems unfruitful to pursue connectionswhich primarily permit only the translation of unsolved topological questions intounsolved group-theoretic questions. Today such thoughts are no longer justifiable.Since generators and defining relations of subgroups of groups presented by gener-ators and relations may be determined, group theory provides a profitable instru-ment of computation for topology, with which several previously inaccessible ques-tions become subject to systematic investigation. Conversely, complexes performa valuable service in making group-theoretic theorems more intuitive and makinggeometric examination fruitful for groups; e.g., planar complexes give information

about the structure of planar discontinuous groups.Accordingly, I have developed the theory of groups presented by generators andrelations as fully as possible, and have favored those fields of topology that bestdemonstrate the connection between groups andcomplexes, and which permit newgroup-theoretic results to be obtained. If, as a result, the topology of 3-dimensionalmanifolds is not explicitly mentioned, nevertheless all methods necessary to attackproblems in this area are presented.

I hope that the reader who wants to acquire a few tools for further work and tolearn positive geometric results in a polished logical structure will also be contentwith the choice of material. In the last four chapters there are numerous results im-mediately accessible to the intuition and which arederived in a logically transparentway from a few simple axioms about points, line segments, and surface simplexes. Ibelieve that Chapters 4 to 6 in particular will present no difficulties in comprehen-

sion. Perhaps it is advisable for the reader to begin with these chapters and, wherenecessary, to refer back to the prerequisites on groups. But Chapters 1 and 2 willalso present little difficulty for most readers. The sections on groups with operators,as well as Chapters 3 and 7 (Chapter 7 is written somewhat tersely), may be omittedthe first time.

Apart from a small sketch in F. LEV ISGeometrische Konfigurationen (Hirzel

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iv CONTENTS

2.13 Characterization of commutative groups . . . . . . . . . . . . . . . . . . . . . . 442.14 Commutative groups with operators . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.15 Characterization of groups with operators . . . . . . . . . . . . . . . . . . . . . 482.16 Divisibility properties ofL-polynomials . . . . . . . . . . . . . . . . . . . . . . . 502.17 Greatest common divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.18 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.19 Factor groups with respect to commutator groups . . . . . . . . . . . . . . . 54

3 Determination of Subgroups 57

3.1 Generators of Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 Generators of the subgroup as special generators of the group . . . . . . 583.3 Properties of the replacement process . . . . . . . . . . . . . . . . . . . . . . . . 603.4 Defining relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.5 SCHREIERSnormalized replacement process . . . . . . . . . . . . . . . . . . . 62

3.6 SCHREIER

S

choice of representatives G. . . . . . . . . . . . . . . . . . . . . . . . 633.7 The relations of the second kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.8 Invariant subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.9 Subgroups of special groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.10 Generators and defining relations of the congruence subgroupUp . . . 703.11 The relations U

uG,SG,S of the groupUp . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.12 Commutator groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.13 The Freiheitssatz (the freeness theorem) . . . . . . . . . . . . . . . . . . . . . . 763.14 Determination of automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4 Line Segment Complexes 83

4.1 The concept of a line segment complex . . . . . . . . . . . . . . . . . . . . . . . 834.2 Orders of points. Regular complexes . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3 The Knigsberg bridge problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.4 Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.5 The connectivity number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 874.6 The fundamental group of a line segment complex . . . . . . . . . . . . . . . 894.7 Coverings of complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.8 Paths and coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 934.9 Simplicity of a covering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.10 Coverings and permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.11 Fundamental domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.12 Regular complexes of even order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.13 Modifications of regular complexes . . . . . . . . . . . . . . . . . . . . . . . . . . 974.14 Invariance of the decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.15 Regular complexes of degree three . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.16 Coverings and permutation groups . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.17 Residue class group diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.18 Regular Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1024.19 Iterated Coverings and Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.20 Transformations into Itself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104


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vi CONTENTS

7.12 Branching Numbers of Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587.13 Automorphisms of Groups of Manifolds . . . . . . . . . . . . . . . . . . . . . . . 159

7.14 The Word Problem for Planar Groups . . . . . . . . . . . . . . . . . . . . . . . . . 1617.15 Word Problems in Planar Group Diagrams . . . . . . . . . . . . . . . . . . . . . 1627.16 Re-entrant Vertices and Critical Subpaths . . . . . . . . . . . . . . . . . . . . . 1637.17 Simple Paths in Planar Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647.18 Planar Group Diagrams and Non-Euclidean Geometry . . . . . . . . . . . . 165


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Introduction

If one wants to investigate surfaces globally, it is often appropriate to divide theminto finitely many surface pieces, bounded by finitely many curve pieces, and iden-tified along these curve pieces in a certain way. This is appropriate for the intu-ition, which can better control pieces in their totality than the idea of a complicatedsurface; appropriate from the standpoint of differential geometry, the methods ofwhich initially give access only to elementary pieces of surfaces and curves; and ap-propriate finally for questions in which the local properties of the surface play no de-cisive role, e.g., when it is to be decided whether two surfaces may be mapped ontoeach other one-to-one and continuously. If we call the boundary relations betweenthe surface pieces, curve pieces, and their endpoints the structure of the decompo-sition then it may be easily shown, e.g., that surfaces possessing decompositions ofthe same structure may be mapped onto each other one-to-one and continuously,and it is possible to prove the even more plausible converse theorem that surfacesthat can be mapped one-to-one can continuously onto each other possess decomo-sitions with the same structure.

However, as soon as the decomposition of a surface is introduced as a tool, it be-

comes an unavoidable question which properties of the surface are expressed in thestructure of a decomposition, or how the structures of different decompositions ofthe same surfaceare related. These questions are the starting point of combinatorialtopology.

In order to answer them, one first asks about the properties of the elementarysurface and curve pieces that bring about the structure. There are a few simple andintuitive facts; say, the fact that a curve piece always has two boundary points, andthat a surface piece always has a boundary curve determined by finitely many curvepieces. By formulating these facts in axioms it is then possible to delimit an area ofgeometrythe combinatorial topology of line segments and surface complexesthe foundation of which is as logically clear as it is intuitively satisfactory. The re-sults of this theory present a noteworthy contrast to the axioms, in that they leadvery quickly to questions that are as difficult to answer as they are easy to ask. The

classical and popular example of this is the four color problem. No situation canshow more clearly that mathematics does not live by logic and intuition alone, andthat any theory requires not only unobjectionable axioms but also fruitful ideas.

Here we must mention the group concept, which very soon proves to be an Ari-adne thread through the labyrinth of complexes, and which we therefore discussfirst, as far as it relevant to combinatorial topology. In this connection, the methods

1


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Chapter 1

Groups

1.1 Definition of the Group Concept

We begin our exposition with an explanation of the group concept and a few simpletheorems on groups, subgroups, factor groups, and isomorphisms of groups.

A classFof elements is called a group when each ordered pair1 F1F2is associatedwith a certain elementF12ofF, in symbols

F1F2 = F12 (F1timesF2equalsF12),

and this linking, or multiplication, satisfies the following rules:

A. 1. If F1, F2, F3are any three elements ofFand if

F1F2 = F12, F2F3 = F23

thenF12F3 = F1F23.

We will write this(F1F2)F3 = F1(F2F3) (1)

for short.

A. 1. is called the associative law, and a multiplication that satisfies it is calledassociative.

Because of (1) we can write the product of an ordered triple of factorsF1, F2, F3as

F1F2F3 = (F1F2)F3 = F1(F2F3)

and, as may be seen byinduction, the product ofnfactors is expressible analogouslyas

F1F2 Fn,

independent of bracketing.

1Notice thatReidemeister denotes an ordered pair simply byjuxtaposing its elements. The moreusualordered pair notation (a,b) is used by himas thenotation for the greatestcommon divisorof integers a,bin Section 1.3. (Translators note.)

3


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4 CHAPTER 1. GROUPS

A. 2. There is an element E inFfor which

E F= F E= F

for any F inF.Such an element is known as an identity element.

There can be only one such element; for, ifEwere a second element of this kind,we must haveE E = Eon the one hand, andE E = E on the other, so thatE= E.

A. 3. For each element F of Fthere is an element X for which

F X= E. (2)

Xis called an element inverse toF.

For an elementXinverse toFwe also have

X F= E. (3)

This is because there is an elementYfor which

X Y= E. (4)

Then multiplication byFgives, on the one hand, that

F(X Y) = F E= F.

While, on the other hand, it follows from (1) and (2) that

F(X Y) = (F X)Y = E Y= Y

and hence Y =F, so the asserted equation (3) follows from (4). It follows furtherthat there is only one inverse element. On the one hand it follows fromF X1 = EandF X2 = Ethat

X2(F X1) =X2E=X2.

And sinceX2F= Eby (3) we have, on the other hand,

X2(F X1) = (X2F)X1 = E X1 =X1,

and thusX1 =X2.IfXis the element inverse toF, then Fis the element inverse toX. We denote

the element inverse toFbyF1, so

(F1)1 = F.

This symbolism may be extended by the following convention. ByF1 we meanFitself. Fn is defined for positive integersn> 1 by induction as

Fn = (Fn1)F.


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1.2. CYCLIC GROUPS 5

ByF0 we mean the identity elementE, byFn (n> 0) we mean (F1)n; it then fol-lows from the associative law and the properties of the inverse that

FnFm = Fm+n, (Fn)m = Fnm

for arbitrarym,n.Fn is therefore the element inverse to Fn. We callFn the nth power of the

elementF. IfF1 and F2are two different elements ofFthen we can construct theelements

Fn11

1 Fn12

2 Fn21

1 Fn22

2 Fnr1

1 Fnr2

2 (5)

by iterated multiplication, which may be called power productsofF1andF2. We call

Fnr2

2 Fnr2

1 Fn22

2 Fn21

1 Fn12

2 Fn11

1 (6)

the power product formally inverse to (5), because one computes that the product

of (5) and (6) is equal toE

. The power products of elementsF

1,F

2, . . . ,F

kmay beconstructed analogously.IfF1F2 = F2F1 for any two elements F1 and F2 ofFthen the group F is called

commutative. If each element ofFmay be written as a power of a fixed element Fthen Fis called a cyclicgroup with generatorF. Since

FnFm = Fn+m = Fm+n = FmFn,

a cyclic group is commutative.With a view towards our objectives we will always assume that the groups un-

der consideration have only a denumerable number of elements. The number ofelements of a group is called itsorder.

1.2 Cyclic GroupsCylic groups are easy to exhibit. E.g., the positive and negative integers and zeroconstitute such a group when one takes addition as the group operation: all num-bers may then be regarded as power products of+1 and1.

The residue classes with respect to a modulus, under addition, constitute an-other example. Ifmis any positive integer we call two integers n1and n2congruentmodulom, denotedn1 n2(modm), if the differencen1 n2is divisible bym. If

n1 n2(modm) and n2 n3(modm)

then alson1 n3(modm).We now understand the residue class [n] to be all numbers congruent to n(mod

m). Obviously, the class [n] is identical with the class [n],

[n] = [n], if n n (modm)

and conversely. For each residue class there is exactly one representativersatisfyingthe inequality

0 r< m.


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6 CHAPTER 1. GROUPS

Thus there aremdifferent residue classes.We define an operation on these residue classes, denoted by the symbol + and

called addition, by[n1] + [n2] = [n1 + n2].

This definition is not contradictory. Namely, if[ni] = [ni]then ni ni (modm)

(i= 1,2) and hencen1 + n

2 n1 + n2(modm),

as one easily verifies, so[n1 + n2] = [n

1 + n

2].

This operation satisfies the group axioms. It is associative because addition of wholenumbers is:

([n1] + [n2] ) + [n3] = [ (n1 + n2) + n3]

= [n1 + (n2 + n3)]

= [n1] + ( [n2] + [n3]).

[0] is the identity element and [n] is the element inverse to [n]. The group is cyclicas well, because all m residue classes result from iterated addition of the residueclass [1].

The multiplication property of cyclic groups is easily seen to lead to the two ex-amples given. If all elements ofFare powersFn (whereFis the generator ofF) theneitherFn is different fromFn as long asnis different fromnin which case mul-tiplication of elements ofFreduces to addition of whole numbersor else there aretwo different exponentsn> nwhich yield equal elements ofF. Then

FnFn

= Fnn

= E

and there is a smallest positive exponent ffor which

Ff = E.

In this caseFn = Fn if and only ifn n (modf). In fact

Fk f = (Ff)k = Ek = E

and soFn+k f =Fn. Conversely, ifFn =Fn (n> n) thenFnn =EHere we musthaven n f, and if

n n = k f+ r (0 r< f)

thenFr = 1, sor= 0. In this case the cyclic group has orderf.

1.3 Multiplication of Residue ClassesWe can construct groups from the residue classes modmin another way by takingthe group operation to be multiplication of residue classes. The product of[n1] and[n2] is defined by

[n1][n2] = [n1n2].


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1.4. GROUPS OF TRANSFORMATIONS 9

thenx = F21(x), x = F32(x)

and hencex = F3(F21(x)) = F32(F1(x))

is the same transformation ofx.NowifFis a family2 of such one-to-one transformations and ifFcontains, along

with each member F , the inverse F1 and, along with each pair F1, F2, their productF21, thenFis obviously a group.

If the domainX consists of finitely many objects

x1, x2, . . ., xm

then a one-to-one onto transformation

xni

= F(xi

) (i= 1,2,..., m)

is called a permutation of the objects xi.If suitable transformations in a group Fwill send anyxto any other, then the

transformation group is calledtransitive.As an example of a transformation group we introduce themodulargroup. The

domain X consists of the complex numbers

x= 1 + i2 with 2 > 0

and the transformations are

x =a x+ b

c x+ d, (1)

wherea,b, c, dare integers with determinant

a d b c= 1.

If

x =ax+ b

cx+ d (2)

is a second such transformation, then the composite transformation

x =ax+ b

cx+ d (3)

has

a = aa+ bc, c = ca+ dc,

b

= a

b+ b

d, d

= c

b+ d

d. (4)

The determinantad bc is equal to the product

(ad bc)(a d b c) = 1.

2Strictly, anonemptyfamily, but Reidemeister always assumes nonempty sets. (Translators note.)


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10 CHAPTER 1. GROUPS

x=d x b

c x+ a (5)

is the transformation inverseto (1). If (1)is the identity transformation then we musthave

c x2 (d a)x b= 0

for allx. It follows thatb= c = d a = 0 and, becausea d = 1, eithera= d = +1ora= d= 1. One sees from this that two transformations defined by the formula(1) are identical if and only if their coefficients a,b, c, dare respectively equal or elserespectively of the same magnitude but oppositely signed.

1.5 Subgroups

In order to penetrate more deeply into the structure of a group Fone considers its

subgroups, i.e., groupsf whose elements all belong to F. In this context the groupoperation for the elements off is the same as that for F. Thus Fitself is a subgroupofF. Each subgroup different from Fitself is called a propersubgroup. One can alsocharacterizesubgroups as follows: a collection fof elements ofFis calleda subgroupwhen

F1F2 = F12

belongs to f along with F1and F2and, along with each elementF, its inverseF1 alsobelongs to f. Obviously the identity element E= F F1 then belongs to Fand, sincethe product of elements in f is naturally associative, f is in fact a group and hence asubgroup ofFaccording to the first definition.

The elements representable as powers of an element Fconstitute a subgroup,because the product of two powers ofFand the inverse of a power are again powersofF. One calls the order of this subgroup the order of the element F. In the example

of the whole numbers these groups consist of all the elements divisible by a givennumber.

One concludes similarly that the power products (5) in Section 1.1, of two or anarbitrary finite or infinite set of elements constitute a group. Thus ifm is any set ofelements ofFone may speak of the subgroup ofFdetermined orgeneratedbym. Itis just the set of all power products of elements ofm.

An important subgroup defined in this way is the commutator groupK1ofF. Bythe commutator ofF1andF2we mean the element

K= F1F2F1

1 F1

2 .

Nowifk1is the set of all commutator elements ofF,thenK1is the group generated byk1. By the commutators of second order, k2, we mean all commutators of an element

ofk1with an element ofF,andbyK2the group so generated, the second commutatorgroup. Commutator groups of higher order may be defined by induction.

The elementsFthat commute with a fixed elementF0, i.e., those for which F0F=F F0, constitute a group. For if

F0F1 = F1F0 and F0F2 = F2F0


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1.6. CONJUGATE SUBGROUPS 11

then alsoF0F1F2 = F1F0F2 = F1F2F0

andF0F

11 = F

11 F1F0F

11 = F

11 F0F1F

11 = F

11 F0.

Similarly, one concludes that the set Z of those elements ofFthat commute with allthe elements ofFconstitute a subgroup. It is called thecenterofF.

It is also easy to construct subgroups of a group of transformations. The set of alltransformations that leave a given element x0fixed, e.g., constitute a subgroup. ForifF1(x0) = x0andF2(x0) = x0then also

F2(F1(x0)) = F21(x0) = x0

andF11 (x0) = x0.

Similarly, the transformations that leave several points xfixed constitute a sub-group.

Iff1and f2are subgroups ofF, then so is the collection f12of all elements thatbelong to both f1and f2. Namely, ifF1andF2are elements that belong to both f1andf2, then F1F2 = F12and F

11 also belong to bothf1 and f2. One concludes similarly

that the intersection of arbitrarily many subgroups is also a subgroup.

1.6 Conjugate Subgroups

Iff is a subgroup ofF,Fruns through all the elements off, andF0is a fixed elementofF, then the elements

F0F F1

0 = F

run through a collection f of elements that also constitute a group. For if

F1F2 = F12

thenF

1F

2 = F0F1F1

0 F0F2F1

0 = F0F12F1

0 = F

12

andF0F1

F10 is the element inverse toF0F F1

0 . We also writeF0fF1

0 for f and call

it asubgroup conjugate tof.Iff is conjugate to f, then f is also conjugate to f. For in fact F10 f

F0is identicalwith f. Iffuis a proper subgroup off, thenF0fuF

10 is a proper subgroup ofF0fF

10 .

Iff and f are two subgroups conjugate to f, and if

f = F0fF10 and f = F10 fF0,

and if both f and f are contained in f, then f, f, f are identical. E.g., iffwere aproper subgroup off then we would also have

F10 fF0 = f


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1.8. RESIDUE CLASSES MODULO SUBGROUPS 15

if there is an elementFfoff such thatF1 = F2Ff. If

F1rF2 (mod f) then also F2rF1 (mod f),because in factF2 = F1F

1f . And if

F1rF2 (mod f) and F2rF3 (mod f)

then alsoF1rF3 (mod f),

because ifF1 = F2Ff and F2 = F3Ffthen F1 = F3Ff Ff =F3Ff. We understand theright-sided residue class4 determined byF1to be the collectionF1f of elements rightcongruent toF1modulo f. IfF1rF2(mod f) then

F1f= F2f,

and conversely. The residue class is determined by any one of its elements, or in

other words: two right-sided residue classes modulo f that have a common elementare identical. Thus the elements ofFare partitioned by a subgroup into disjointresidue classes. Asystem of representativesof these residue classes is a set rof ele-mentsRof the following kind: ifFis any element ofFthen there is an element Rofrsuch that

FrR (mod f),

and ifR1 and R2 are two elements ofrthen R1 is not right congruent to R2. Theclasses Rf then yield all the residue classes, without repetitions, as Rruns through r.

We can define left congruence analogously to right congruence. We say

F1l F2 (mod f)

ifF1 = FfF2, where Ffbelongs to f. We analogously define the left-sided residue

classes and a system of representatives r of them. We will show:ifris a fullsystem ofrepresentatives for the right-sided residue classes, thenr1, the collection of inversesto

the R inr, is a left-sidedsystemof representatives. Namely, ifFis an arbitrary elementandF1 = R Ff, thenF= F

1f R

1, so

FlR1 (mod f).

Further, ifR11 lR

12 , so R

11 = FfR

12 ,

thenR2rR1(mod f).If the number of right-sided residue classes modulo f is a finite number n, then

this shows that the number of left-sided residue classes is alson.RfR1 runs through all subgroups conjugate to f asRruns through the set r. For

ifF0is an element ofFandF0 = R0Ff then

F0fF1

0 = R0FffF1

f R10 = R0fR

10 .

4This residue class if of course what we now call acosetof the subgroup f. However, I have thought itbest to retain the term residue class (in German, Restklasse) to reflect Reidemeisters view of cosets asgeneralizations of residue classes in number theory. (Translators note.)


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1.9 Residue Classes modulo Congruence Subgroups of

the Modular GroupAs an example, we determine the left-sided residue classes modulo the subgroupUnof the modular group defined in Section 1.7, fornequal to a prime numberp.

If we denote the transformation

x = 1

x+ k

byGk, then the identity transformation Eand the Gk (k =0,1,...,p 1) form afull system of representatives for theUpM, whereMdenotes an arbitrary modularsubstitution. Namely, for eachMthat does not belong to Up(that is,c 0 (modp))we may determine a substitution Uin Up

x

=

ax+ b

cx+ d,

soc 0 (modp), and a Gkso that

UGk= M.

If the substitutionM G1k is given by

x =ax+ b

cx+ d,

then by (4) of Section 1.4c = k c d.

Thuskmust satisfy the congruence

k c d 0 (modp)

and then by Section 1.3 the residue class [k]is uniquely determined, because c isassumed to be relatively prime topand hencekis also, because

0 k< p.

But then the coefficients ofUare likewise determined by

U= MG1k .

1.10 Factor Groups

The definition of addition of residue classes can also be generalized under the hy-pothesis that the modulus f is an invariant subgroup ofF. One sees first of all that:iff is an invariant subgroup and if

F1rF2 (mod f)


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element. In this case the elements Fthat form the identity element ofF/f are justthose that make up f.

Conversely, ifI(F) = F is a homomorphism, then the identity element ofFmustbe associated with the identity ofF, i.e., I (E) = E, because indeed

I(E)I(F) = I(F)I(E) = I(F).

Consequently, I (F) is inverse to I (F1), because

I(F)I(F1) = I(E) = E.

We now take f to be the collection of elementsFofFfor which I(F) = E. They forma group, because if

I(F1) = I(F2) = E

then alsoI(F1F2) = E

.

And sinceI(F1)I(F

11 ) = E

I(F11 ) = I(F

11 )

and, on the other hand,

I(F1)I(F1

1 ) = I(F1F1

1 ) = E,

F11 belongs to this collection along withF1. Finally, f is an invariant subgroup ofF.Indeed,

I(F F1F1) = I(F)EI(F1) = E.

It follows easily from this that Iassociates all elements of a residue classFfwiththe sameF and hence it realizes an isomorphism between F/f and F.

From Section 1.2 it follows that cyclic groups of the same order are isomorphic.

1.12 Automorphisms

A one-to-one onto transformation of a group Finto itself,

F = I(F),

is called an autoisomorphism, or simplyautomorphism, when

I(F1)I(F2) = I(F1F2).

The automorphisms of a groupFconstitute a group. For the identity mapping isobviously an automorphism, likewise the inverse I1 of an automorphism. Namely,ifF1 F

2 = F

3and F

i = I(Fi), so that I

1(Fi ) = Fiand I(F1)I(F2) = I(F3) then

I(F1F2) = I(F3)


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1.12. AUTOMORPHISMS 19

because I is an automorphism. Thus F1F2= F3because Iis one-to-one, and sinceFi= I

1(Fi ) it follows that

I1(F1)I

1(F2) = I1(F3).

Further, ifI 1(F) = F and I2(F) = F are automorphisms then

F = I2(I1(F)) = I21(F)

is likewise an automorphism, because

I21(F1)I21(F2) = I2(I1(F1)I1(F2))

= I2(I1(F1F2))

= I21(F1F2).

Finally, the product is associative because it is the composition of functions.It is easy to exhibit particular automorphisms. IfF0is a fixed element ofFandF

runs through all the elements ofF, then

F = F0F F1

0

is a one-to-one onto transformation of group elements because

F10 FF0 = F

is the inverse mapping, and also

F0F1F1

0 F0F2F1

0 = F0F1F2F1

0 .

Such a mapping is called aninner automorphism. IfF0runs through all elements ofFwe obtain the totality of inner automorphisms ofF. They constitute a group thatis homomorphic to Fitself. The product of two inner automorphisms

F = F1F F1

1 and F = F2F

F12

is another, namely,F = F2F1F F

11 F

12 = F21F F

121 .

In order to ascertain whether the homomorphism from Fto the group of its innerautomorphisms is one-to-one we must establish which inner automorphisms cor-respond to the identity mappingbut these are just those defined by elements be-longing to the center ofF. The group of inner automorphisms is therefore isomor-phic to the factor group F/Z.

The inner automorphisms are an invariant subgroup of all the automorphisms.Namely, ifA(F) = F

is an arbitrary automorphism, and

I(F) = F = F0F F1

0


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is an inner automorphism, then

A(I(F)) = A(F0)A(F)A(F10 ) = F0 FF0 1.

Thus if we set

I(F) = F= F0F

F01

= F0 A(F)F

01

then

A(I(F)) = I (A(F))

so

A I= I A or AI A1 = I.

Iff is an invariant subgroup ofFandF0is any element ofF, then the mapping

F0F F

1

0 = F

is an automorphism off. IfF0itself belongs to f, then it is an inner automorphism.The totality of automorphisms induced by elementsF0ofFconstitute a subgroup ofall the automorphisms off. The elements of a residue classF0fmodulo f correspondto automorphisms resulting from multiplication by inner automorphisms.

One can see from these remarks that any group can be embedded as an invariantsubgroup of a larger group. We want to formulate the situation as follows: givena groupf and its product operation, together with a system rof representatives ofresidue classes ofFmodulo f, one then knows that each element ofFmay be writtenas a product R F, where Ris fromrand F is fromf. In order to extend the groupproduct to all ofF, i.e, to know the value of the product

R1F1R2F2 = R1R2R1

1 F1R2F2,

we must first know the automorphisms off corresponding to the elements Randalso, for any two elements R1, R2, the productR12F12. Then the group Fitself will beknown.

1.13 Groups with Operators

When a groupFwith a cyclic group of automorphisms An is given5 we can makethe structure connecting these two domains of elements clearer by means of a newsymbolism, which is particularly convenient in the case of a commutative group F.So we will assume that F is commutative. ByFx we will mean the element A(F)

and byFxn

the element An(F) (n= 0, 1, 2,. . .). For any integer an, Fanxn

means(Fan)x

n.If

f(x) = anxn + an+1x

n+1 + + an+mxn+m

5That is, consisting of the powers of an automorphism A. (Translators note.)


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1.13. GROUPS WITH OPERATORS 21

is an L-polynomial,6 with integral coefficientsai, then byFf(x) we mean the ele-ment

Ff(x) = Fanxn Fan+1xn+

1 Fan+mxn+

m

One can compute in this extended domain of exponents as in the original do-main of integers. We call two polynomials equal if they are convertible into eachother by deletion or insertion of termsaixi withai =0. If f(x)and g(x)are twopolynomials and nis the lowest, and n+ mthe highest exponent of an xi appearingin f andgalong withaior bi= 0, then

f(x) = anxn + an+1x

n+1 + + an+mxn+m

g(x) = bnxn + bn+1x

n+1 + + bn+mxn+m.

As usual, we understand the sum off(x) andg(x) to be the polynomial

f(x) + g(x) =

n+mi=n

(ai+ bi)xi

.

This addition operation satisfies the laws of a commutative group, because the in-tegers under addition are such a group. The polynomial f = 0 plays the role of theidentity element.

We understand the product off(x) and b xl to be the polynomial

anb xn+l + an+1b x

n+1+l + + an+mb xn+m+l

and as usual we understand the product off(x) andg(x) to be the polynomial

f g= f(x)bnxn +f(x)bn+1x

n+1 + +f(x)bn+mxn+m.

This multiplication is associative and commutative; f(x) = 1 is the identity element.

However, an inverse element does not exist in general; e.g., the polynomial f(x) = ahas no inverse when a = 1, because the coefficients of all products a f(x)aredivisible bya. The multiplication and addition are further related by the distributivelaw:

(f1(x) +f2(x))g(x) =f1(x)g(x) +f2(x)g(x).

Iff(x) andg(x) are two polynomials, both nonzero, and ifanxn andbmxm are thelowest-order terms appearing in f(x)and g(x)with an =0 and bm =0, then thelowest-order term appearing in f(x)g(x)is anbmxn+m. It follows from this that iff(x)g(x) = 0 then at least one of the factors f(x) org(x) equals zero. Thus the poly-nomials constitute an integral domain.7

If the smallest exponent in a polynomial f(x) is greater than or equal to zero thenf(x) is an ordinary8 polynomial in x. For any ordinary polynomial

f(x) = anxn + an+1xn+1 + + an+mxn+m

6TheLpresumably stands for Laurent, since these polynomials can have terms with negative expo-nent. (Translators note.)

7See a textbook of algebra, e.g., H. HAS SE Hhere Algebra, Band 1, Sammlung Gschen.8Reidemeister calls such a polynomial entire, following the terminology of complex analysis. But it

seems harmless, and clearer, to call such polynomials ordinary. (Translators note.)


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withan= 0xnf(x) = an + an+1x+ + an+mx

m

is an ordinary polynomial with nonzero constant term.If we now consider f(x)and g(x)as exponents of group elements, it turns out

thatFf(x)Fg(x) = Ff(x)+g(x).

This follows easily from the commutativity of the group and the definition ofFf(x)

andf(x) + g(x). Further,

(Ff(x))g(x) = Ff(x)g(x),

because(Ff(x))g(x) = (Ff(x))bnx

n

(Ff(x))bn+1xn+1

(Ff(x))bn+mxn+m

.

Then, on the one hand,

(Ff(x))bi = Fbif(x)

since(Ff(x))bi = Ff(x)Ff(x) Ff(x) with bifactors

for positive bi, and for negative bi

(Ff(x))bi =

(Ff(x))1bi

=

Ff(x)bi

.

While, on the other hand,

(Ff(x))xi

=

Fanxn

Fan+1xn+1

Fan+mxn+mxi

= FanxnxiFan+1xn+1xi Fan+mxn+mxi= (Fan)x

n+i

(Fan+1 )xn+1+i

(Fan+m)xn+m+i

= Fxif(x).

Consequently, (Ff(x))bixi = Fbixif(x) and hence

(Ff(x))g(x) = Fbnxnf(x) Fbn+1x

n+1f(x) Fbn+mxn+mf(x)

= Fbnxnf(x)+bn+1xn+1++bn+mxn+mf(x)

= Ff(x)g(x).

Thus one can compute with the formally introduced L-polynomials as expo-nents just as with integral exponents.

1.14 Groups and Transformation Groups

We call a transformation group Tthat is homomorphic to an arbitrary group Farepresentation ofF, and we further examine the different representations of a group.


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1.15. THE GROUPOID 23

We take as our domain X of objects the right-sided residue classes modulo asubgroup f, so x= Rf, and define

F(x) = F Rf= x

to be the transformation of this domain corresponding to the group elementF.This mapping is one-to-one and onto, because F1 yields the inverse mapping.

The transformations that carry the element x =f into itself are exactly those thatcorrespond to elements off. The transformations that carryRf to itself correspondto the elements of the group RfR1 conjugate to f. We can now easily give a criterionfor isomorphism between Fand the group just defined.

Those elements that correspond to transformations leaving allxfixed must there-fore belong to the intersection D of the groups RfR1 conjugate to F. The group Tistherefore isomorphic to the factor group F/D.

Iff is an invariant subgroup, then the transformations that correspond to f, and

thus carry the element x =fto itself, also carry all the remaining xto themselves,because D in this case is equal to f. The transformation group is then simply transi-

tive.Conversely, given any simply transitive group of transformations isomorphic to

F, an arbitrary element x0, and fx0 the subgroup of transformations that leavex0fixed, the transformations that carryx0to xconstitute a residue class modulofx0 .Namely, ifRxcarries the element x0 to x, so also do the transformations in Rxfx0 ,and ifR is any transformation that carriesx0to x, thenR1x R

carries the elementx0to itself and it therefore belongs to fx0 .

If we associate with each group element

F(x) = x

the transformation

F(Rxfx0 ) = F Rxfx0

in the domain of residue classes, then

F Rxfx0 = Rxfx0 .

Thus the new transformation group is simply the original one with renaming of theobjects transformed. A representation of a groupFby a transitive transformationgroup is an isomorphism, by the remarks above, if and only if the domain of objectscan be viewed as a system of right-sided residue classes modulo a subgroup f, wherethe intersection offwith its conjugate subgroups is the identity.

1.15 The Groupoid

For many topological questions a generalization of the group concept, the groupoid,9

is a useful auxiliary.A collectionG of elements G with a product G1G2 = G3is called a groupoidwhen

the following conditions are satisfied.

9H. BRANDT, Math. Ann.96, 360.


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Chapter 2

Free Groups and their Factor

Groups

2.1 Generators and Defining Relations

The groups that appear in combinatorial topology are defined in a way that itselfhas a combinatorial character. The peculiar difficulties of topology can be betterappreciated when one has at hand the analogous problems of group theory, whichwe are about to present.

IfFis any group and m is a class of elements from which all elements ofFmaybe constructed as power products, thenm is called a system of generatorsfor thegroupF. Thus a system of generators for the integers is just 1, and for an additiveresidue class group it is the class [1]. These examples already draw attention to the

fact that formally different power products can yield the same group element. Ifone wants to be able to derive the product of elements ofFfrom the product ofpower products, then one must be able to decide which power products representequal group elements. This reduces to the question of which products represent theidentity element.1

We call each productR(m)of elements ofm that equals the identity a relation,and call the totality of relations R. Now ifPis any power product andRis any rela-tion, then obviouslyPand P Rare the same group element

P= PR.

Conversely, ifP1and P2aretwo power products that denote the same group element,and ifP11 is the product formally inverse toP1, thenP

11 P2is a relation R

, and the

power product P2results from the product P1Rby deletion of adjacent factorsF F1.We thus obtain all representations of the elementPin the form P Rwhen R runsthrough the class R, if we also include those products that result from deletion offormally inverse adjacent factors ofPR.

1The so-calledword problemfor the group F. (Translators note.)

25


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26 CHAPTER 2. FREE GROUPS AND THEIR FACTOR GROUPS

The power products ofR have the following properties.IfR =P1P2belongs toR, so doesR =P1F F1P2, and conversely, ifR belongs

toR so does R. IfRbelongs toR, so does the formal inverse R1. IfPis an ar-bitrary power product, P1 its formal inverse, and ifRbelongs to R, then P R P1

also belongs to R. IfR1 and R2 belong to R, then the product R1R2 also belongstoR. By means of these four processes, consequence relations may be derivedfrom relations originally given. We call a class rof defining relations from which allrelations in R may be derived by the four processes a system of defining relations.With a class m of generators and a class rof defining relations the product law isobviously defined for all elements ofF, hence the name defining relations. Estab-lishing generators and defining relations for groups given in other ways is a far fromtrivial problem.2

Just as for groups, one can speak ofgeneratorsfor a groupoid. We will assumethat one can find, from generators Si(i= 1,2,..., m) of a groupoid Gwith identitiesEi (i =1,2,...,n), a system of generators Tiof the group G0 of elements doublyassociated with the identityE0.

LetAi(i= 1,2.. . , n) be a system of elements with left-sided identityE0and right-sided identities including allEi(i= 1,2.. . , n). Further, letA0 = E0. Now ifSihas theleft identityEliand the right identityErithen the element

Ti=AliSiA1ri

(1)

may be called the generator ofG0associated withSi. In fact, the Ti (i =1,2.. . , n)constitute a system of generators for G0. Namely, if

S11

S22

Saa

(2)

is any element ofG0, then the left identity ofS11 and the right identity ofS

aais the

identityE0and, further, the right identity ofSiiis identical with the left identity ofSi+1i+1 . The product

T11

T22

Taa

that results from (2) when Siis replaced byTimay be converted into (2) by means ofequation (1) and cancellation of formally inverse factors Si.

2.2 Free Groups

Instead of starting with a group and constructing generators and defining relationsfor it, we will now proceed from a class m of symbols, define the power products ofthese symbols, take an arbitrary systemrof these power products, and show thatthere is a group Fthat has the symbols in m as generators and the products in rasdefining relations. For this purpose we first explain relation-free groups, or simplyfree groups with n generators. Let

S+11 ,S+12 , . . . ,S

+1n ,S

11 ,S

12 , . . . ,S

1n

2Cf. Sections 2.9 and 3.1 and, e.g., J. NIELSEN,Kgl. Dan. Vid. Selsk., Math. fys. Med. V, 12 (1924).


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whereWis a short word. Wmay be called the kernel ofW. By{{W}}we will meanall those words that have a kernel in {W}. All words in{{W}}obviously correspond

to elements that are transforms of each other.One now sees that, along with W, the word that results from SiW

Si by reduc-tion also belongs to {{W}}, and from this it follows that the elements belonging to{{W}}are all the transforms of this element.

2.5 Groups with Arbitrary Relations

We now construct a group with generators

S1, S2, . .. , Sn

and defining relations

R1(S), R2(S), . .. , Rm(S),where the Riare any words inthe Si. We first construct the free groupS determinedby the Si. TheRiare extended by adjoining all L RiL1, whereLis an arbitrary ele-ment ofS, and we construct the subgroup R ofS consisting of all power productsof theRiand their transforms L RiL1. This is obviously an invariant subgroup ofS. Thus we can construct the factor group F=S/R ofS byR by Section 1.10. Weclaim that the residue classes

S1R, S2R, . .. , SnR

generate this group and that the Riyield a system of defining relations for F in thegeneratorsSRwhenSiis replaced bySiR. The productsRi(SR) that result in thisway are certainly relations, for it follows from

S1i RS1k R= S

1i S

1k R

thatRi(SR) = Ri(S)R=R.

Conversely, ifR(SR) is any relation in the group S/R thenR(S)R=R, soR(S) mustbelong toR; i.e., R(S) may be written as a power product of the Ri(S) and their trans-formsLRiL1. Thus theRi(SR) really are a system of defining relations forS/R.

Since each word in the Sicorresponds to a well-defined element of the groupS/R, we can regard it as a notation for this element and, e.g., speak of the elementSiof the groupS/R=Fand hence call Fthe group with generatorsSi(i= 1,2,..., n)and defining relations Rk(k= 1,2,..., m).6 On the other hand,IfF is a group withthe generatorsS

i (i= 1,2,..., n) and defining relations R

k(S) (k= 1,2,..., m), thenF

is isomorphic to a factor group of the free group with n free generators.

IfLandMare arbitrary power products fromS, andRis a power product fromR, then the element LR Mis equal to LM in F. For LR Mis in fact equal to LMM1RM.

6O. SCHREIER,Hamb. Abhdl.5(1927) 161.


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First we give a process that associates with each reduced wordWof the form (1)from Section 2.2 a unique reduced word |W|, equivalent in F. We setW1 = S

11and

|W1| = Sr11 , where1 r1(moda1 ), 0 r1< a1 ,W2 = S11S22 and|W2| = Sr21 where1 + 2 r2(moda1 ), 0 r2< a1 , when1 = 2, and|W2| = S

r11S

r22 where2 r2

(moda2 ), 0 r2 < a2when 1=2.In general, letWi= S

11S

22 S

iiand|Wi| = W

iS

ri. Ifi+1 = , let|Wi+1| = W

iS

r

,

whereri+ i+1 r (moda), 0 r


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2.7. THE FREE PRODUCT OF GROUPS 33

may be easily reduced to the case treated above.9 HereSa11 = Sa22 , from which it

follows that the element Sa1 commutes with all elements ofF, because

S1Sa11 = S

a11 S1

andS2S

a11 = S2S

a22 = S

a22 S2 = S

a11 S2,

so Sa1 commutes with all power products of the Si. Each element may then be con-verted into a reduced word of the form

Sr111 S

r212 S

r1m1 S

r2m2 S

ka11 with 0 ri l< ai.

One proves quite analogously as for the groups with defining relations (2) that eachword is representable in only one way as a reduced word. From the solution of theword problem10 one easily obtains that the subgroup ofF generated bySa1 is the

center ofF.

2.7 The free product of groups

The methods of Section 2.2 may be extended without difficulty to the so-called freeproduct11 of groups. Let G1andG2 be two groups with the elementsG1i andG2irespectively. From these elements we construct words

W= G1G2 Gn,

where the Giare any elements ofG1or G2different from the identity. Thus

Gi= Gkili (ki= 1 or 2).

By an elementary expansion of this word Wwe mean the insertion of a word Gi1Gi2that equals the identity when regarded as a product in Gi, or replacement of a letterGkiliby two, G

kili

Gkili, the product of which equals Gkili in Gki. By a reduction wemean the reverse process.

Again the words Wmay be divided into equivalence classes [W] and the productdefined as in Section 2.2 by

[W1][W2] = [W1W2].

The resulting group G = G1G2is called the free product12 ofG1andG2. One candefine the free product of any number of groups by iteration.

The free group with nfree generatorsSi is the free product of the ninfinite cyclicgroups generated by the Si. The groups (4) of Section 2.6 are the free products ofn

9M. DEH N, Math. Ann.75(1915) 402 and O. SCHREIER loc.cit.10Further solutions of word problems are found in Section 7.14. See also W. MAGN US, Math. Ann.105

(1931) 52 and106(1932) 295; E. ARTIN, Hamb. Abhdl.4 (1925) 47; K. REIDEMEISTER,ibid.6(1928) 56; M.DEHN , Math. Ann.72(1912) 41.

11O. SCHREIER,Hamb. Abhdl.5(1927) 16.12Reidemeister uses the notationG1G2, which I have dropped because of its potential for confusion

with the direct product. (Translators note.)


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finite cyclic groups generated by the Siwith Sai = 1. This construction is importantfor theword problem, because onecan obviously solve the problem in a free product

G as soon as it is solved in the original groups Gi. This is because the reduced word|W|may be defined analogously as in Section 2.3a word is called reduced whenany two neighboring factorsGi,Gi+1 do not belong to the same groupand it isthen demonstrable that each class of reduced words contains only one in reducedform.

If

S1k (k= 1,2.. . , n1), S2k (k= 1,2,..., n2)

are systems of generators for the groupsG1andG2, and

R1l(S1k) (l= 1,2,..., m1), R2l(S2k) (k= 1,2,..., m2)

are the respective sets of defining relations ofG1,G2, then all the Si kand all the Ri ltogether constitute a system of generators and defining relations for the free product

G. It is clear that theRi lare satisfied in G. On the other hand, one can carry outexpansion and reduction of the word W, where Giis now viewed as a power productof the Si k, on the basis of the relationsRi l, because these operations take place onlybetween elements of the same group Gi. Hence theRi l(Si k) are in fact the definingrelations ofG.

It follows conversely that, given a group with generatorsS1kandS2kand a sys-tem of defining relations that can be divided into two classes R1l andR2l, in whichonly the Si lappear in theRi k, then the group in question is the free product of thesubgroups generated by the S1kand the S2k.

The concept of the free product may be extended in thefollowing way. ThegroupG1may possess a subgroupU1that is isomorphic to a subgroupU2ofG2. Let I(U1) =U2be a specific isomorphism between the Ui. Under these assumptions we add thefollowing process to expansion and reduction of words (1): ifGiis an element ofU1

orU2then Gimay be replaced byI(Gi) fromU2or I1

(Gi) fromU1.Classification of words can again be carried out and it leads, again with the helpof equation (2), to the definition of a group G, which may be called the free productofG1andG2with the subgroupsU1andU2amalgamated.

A uniquely determined normal form may now be produced as follows: in thegroupsGiwe choose a system of representatives for the residue classes modulo U1,sayU1N1k, and modulo U2, sayU2N2k, and then one can put each word Win theform

U N1N2 Nn,

where Ubelongs toU1, theNlare certain representativesNi k, and two neighboringNi, Ni+1do not belong to the same group Gl.

From this one can solve the word problem in G if one can give each element Giin Glthe representationUiNi k(i= 1,2). IfS1k (k= 1,2,..., u) are the generators ofU1, S1k (k= 1,2,..., n1) the generators ofG1, and analogously ifS2k (k= 1,2,..., u)are the generators ofU2, andS2k (k =1,2,..., n2) are those ofG2; and ifR1l(S1k)(l= 1,2,..., m1) and R2l(S2k) (l= 1,2,..., m2) are the defining relations ofG1and G2respectively; and if finally the mapping

I(S1k) = S2k (k= 1,2,..., u)


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2.8. A TRANSFORMATION PROBLEM 35

is an isomorphism betweenU1andU2;thentheSi k(k= 1,2,..., ni;i= 1,2), togetherwith the relationsRi l(S), (l = 1,2,..., ni;i = 1,2) andS1k = S2k, (k= 1,2,..., u) are

generators and relations for the free product with amalgamated subgroup, as onemay prove analogously with the theorem on the free product itself.

2.8 A transformation problem

The groups treated in Section 2.6 admit an easy solution of the transformation prob-lem. However, for what follows we will need only the special case of the relations13

R1 = S31, R2 = S

22.

We alter the normal form of Section 2.6 by always writing S11 in place ofS21. Ifi=

1 then each element different from the identity can be brought into one of thefollowing reduced forms

W= S11S2S

21S2 S

m1 ; W S2; S2W; S2W S2; S2 (1)

ByW1we mean the power product formally inverse toW,

W1 = Sm1 S2S

21 S2S

11 ,

and similarly for the other reduced products. Now for the solution of the transfor-mation problem we remark that the first and last factors of a product (1) are eithera) formally inverse to each other or b) not.

In the first case a) we can put the product in the form

H= LHL1

whereLand L1 are formally inverse to each other and where the kernel H of theproductbeginsand ends with factors that arenot formally inverse to each other. ThekernelH has the formWof (1) with1 = mwhen Lends withS2, but it containsonly one factor S2whenLends with S1. In the second case b) the product Hhas oneof the forms S2,Wwith 1 = m,W S2, or S2W.

We will call the products S1, S2,W S2, and S2Wshort words of the first kind. Theproducts S1,S2, andWwith1 = mwill be called short words of the second kind.Each element has a transformed product that is a short word of the first kind; this isbecause it is either a short word of the first or second kind or else it has a kernel thatis a short word of the second kind, and a short word of the second kind becomes ashort word of the first kind by transformation with an S1and reduction.

We now letKdenote a short word of the first kind and let{K}1denote the col-lection of products that result from K by cyclic interchange of factors. By {K}

2we mean the collection of short words of the second kind that result from a wordW S2 = S

11S2 S2out of{K}1by the process

S11 W S2S

1 = S11 S2S

21S2 S

m1 S2S

11 ,

13K. REIDEMEISTER,Hamb. Abhdl.8(1930), 187.


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as well as those short words of the first kind from {K}1that are also of the secondkind. By{K}3we mean all those words Hthat have a kernel H in{K}2. Finally, let

{K}denote the totality of elements from the classes{K}i(i= 1,2,3).Each element obviously belongs to exactly one class {K}. Further, it is clear on

the one hand that any two products in {K}are convertible into each other by trans-formation and reduction, and hence they denote transforms of each other in ourgroup, while on the other hand, ifHis any wordin {K} then S1HS

1 and S2HS2yield

other words in {K}by reduction. One verifies this by considering the cases whereH lies in {K}1, { K}2, or{ K}3. It follows in general that M H M1 yields a word, byreduction, that lies in the same class{K}asH.

Now, on the one hand, we can decide whether two reduced products belong tothe same class{K}, and on the other hand each element of our group correspondsto a unique reduced product, so the transformation problem is solved.

One moreremark about the powers ofan element H. IfHbelongs to a class {K}i,then each powerHk belongs to a class{K}iwith the same indexi.

2.9 Generators and relations for the modular group

The modular group defined in Section 1.4 is isomorphic to the group discussed inthe previous section. Thus we have solved the transformation problem for the mod-ular group.

One can of course also solve the transformation problem by proceeding fromthe arithmetic representation of the substitutions and asking what conditions thecoefficients

a,b, c, d and a,b, c,d

must satisfy for the associated substitutions to be transformable into each other in

the modular group. However, this way is much more difficult. It is connected withthe question of when two binary quadratic forms

Ax2 + Bx y+ C y2 and Ax2 + Bxy+ Cy2

are equivalent, i.e., when there there are integers

a,b,c, d with a d b c= 1

such that the unprimed form goes to the primed form when x,yare replaced by

x= a x+ b y, y = c x+ d y.

We now apply ourselves to the proof thatthe modular group is generated by twoelements S1and S2which satisfy the relations

R1 = S31, R2 = S

22

and no others independent of them.We letTbe the substitution

x = x+ 1,


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2.9. GENERATORS AND RELATIONS FOR THE MODULAR GROUP 37

so thatTn is the substitutionx = x+ n.

BySwe mean

x = 1

x.

IfAis the substitution

x =a x+ b

c x+ d with |b| |d|> 0,

andA = TnA

corresponds to the substitution

x

=

ax+ b

cx+ d,

thenb = b+ nd

and hence by suitable choice ofnone can obtain

|b| < |d| |b|.

If0< |b|< |d|

then the substitution SAor

x =c x+ d

a x bsatisfies the previous condition. Hence it follows by induction that: for each trans-formationAthere is a power product

M= STn1STn2S TnmS

(and equal 0 or 1) such that, in the transformation

x =a x+ b

c x+ d

corresponding toM A, we must haved= 0. It must then be that b c= 1, i.e.,

x = 1

x + a,

and this isTaS. Consequently, SandTare generators of the modular group.Now we setS1 = TS, S2 = Sand confirm thatS22 = 1. Further,S1corresponds to

the substitution

x = 1

x + 1 =

x 1

x .


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S21corresponds to

x = 1 1

x+ 1

+ 1 = xx 1

+ 1 = 1x+ 1

,

so that S31 = 1. SinceT= S1S12 , S1and S2are also generators of the modular group.

They satisfy the two given relations, and it remains only to prove that they satisfyno other relations apart from consequences ofR1and R2. We will show that, if onecomputes the substitution

x =a x+ b

c x+ d

for a reduced word (1) from Section 2.8 in which one replaces the Siby the corre-sponding modular substitutions, then it is never the identity substitution. It sufficesto prove this for words of the formW S2since, by Section 2.8, each element of thegroup may be converted into a wordW S2by transformation with S1or S2.

For the proof we convert W S2back to a certain power product ofSand T. Namely,we combine all neighboring elements S1S2into powers (S1S2)i and likewise the el-ements S11 S2into powers (S

11 S2)

k and then set

(S1S2)i = Ti, (S11 S2)

k = STkS, (i,k> 0).

One sees that this gives a productin Sand Tin which the exponents have alternatingsigns. But it is easy to see that such an element is never the identity substitution bycomputing the coefficients of the corresponding modular substitution.14

Another method of detemining generators and defining relations for the mod-ular group consists in the construction of its fundamental domain in the complexnumber plane.15

2.10 A theorem of TIETZE

It is clear that a group may be defined in various ways by generators and relations. If

S1, S2, . .. , Sm

is a system of generators for a group Fand the set rof products

R1(S), R2(S), . . ., Rr(S)

in the Siis a system of defining relations, and ifRr+1(S) is any consequence of theserelations, then, e.g., the set that results from rby addition ofRr+1is also a systemof defining relations. If, on the other hand, Rr(S) is a consequence ofR1(S),R2(S) ...,

Rr1(S) then the latter set is also a system of defining relations for F.Further, ifTis a letter denoting any power product of the Si,

T= T(S),

14Cf. DIRICHLET-DEDEKIND, Vorlesungen ber Zahlentheorie, 2nd edition, 1871, 81.15Cf. a textbook on function theory, e.g., that of B IEBERBACH,vol. II.


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2.10. A THEOREM OFTIETZE 39

thenRr+1 = T(S)T

1

is a relation, andS1, S2, . .. , Sm, T

is a system of generators and, as we will show,

R1(S), R2(S), . .. , Rr(S), Rr+1(S, T)

is a system of defining relations. This is because each relation containing only theSis a consequence of theRi(i= 1,2,..., r) and, using the relation Rr+1, each powerproduct containing a factorTmay be converted into one in the Salone. Namely, if

F=A(S)T B(S, T)

then

F=A(S)T T1(S)T(S)B(S, T)

=A(S)R1r+1T(S)B(S, T)

=A(S)T(S)B(S,T)

and the latter product contains one Tfactor fewer than Fdoes. In this way thefactors T ( = 1) may be removed successively.

On the other hand, ifSmis representable as a power product ofS1,S2, . . . ,Sm1thenS1,S2, . . . ,Sm1 obviously constitute a system of generators. One can succes-sively eliminate Smfrom all power products. Further, if the defining relationsR1,R2,. . . , Rr1contain only the generators S1,S2, . . . ,Sm1and if

Rr= Sm(S1,S2, . . . ,Sm1)S1m

then theRi(i= 1,2,..., r 1) constitute a system of defining relations in the gener-ators

S1, S2, . .. , Sm1.

This is because the group defined by the

Si(i= 1,2,..., m 1), Rk(k= 1,2,..., r 1)

is, as we saw above, identical with that defined by

Si(i= 1,2,..., m), Rk(k= 1,2,..., r).

We now have an important theorem (of TIETZE16) that any two systems of genera-tors and defining relations for the same group are always convertible to each other by

successive applications of the transformations above.

16H. TIETZE, Mon. f. Math. u. Phys.19, p. 1.


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Let

S1,S2, . . . ,Sm; R1(S),R2(S), . . . ,Rr(S) (1)S1,S

2, . . . ,S

m; R

1(S

), R2(S), . . . ,Rr(S

) (2)

be two systems of generators and defining relations for the same group F. TheSkmust then be expressible in terms of the Siand, conversely, the Siin terms of the Sk.If

Sk= Sk(S); Si= Si(S

)

then we setUk(S,S

) = Sk(S)S1k ; Vi(S,S

) = Si(S)S1i .

Obviously the Si,Skare a system of generators and the relations

Rl(S), Uk(S,S) (3)

on the one hand, as well as the relationsRl(S

), Vi(S,S) (4)

on the other, are systems of defining relations for Fthat result from (1) and (2) re-spectively by successive addition of the respective generators Skand Siwith the re-spective relations UkandVi.

But now the relations (4) must be consequences of (3), because the relations areindeed relations in theSi,Sk. Similarly, the relations (3) are consequences of (4).Hence by addition of consequence relations we can extend both systems, (3) and(4), to the same system

Si,Sk; Rl(S

), Rl(S),Uk(S,S),Vi(S,S

), (5)

and hence convert the system (1) to the system (2) by a sequence of the transforma-

tions described.One can apply this theorem to a purely combinatorial characterization of the

properties of a groupgiven by generatorsSiand defining relations Rk. Each prop-erty of a system of generators Siand relations Rkthat is invariant under the abovetransformations of the Siand theRkis a property of the group Fdefined bySi,Rk.This is because such a property holds for all presentations of the group by genera-tors and relations and hence it is a property of the group itself. Despite this simpleconnection between different presentations of the same group it is in general notpossible to decide whether two groups presented by generators and relations areisomorphic to each other.17 One also cannot decide whether such a group is a free

17This remarkable claimwas first madeby Tietze (1908) in the paper cited above. At the time when Rei-demeister wrote, a precise concept of algorithmformalizing what it means to decidewas still a fewyears away from being published. It first appeared in publications of Church, Post, and most convinc-

ingly by Turing in 1936. Another two decades elapsed before Adyan and Rabin proved that the isomor-phism problem is algorithmically unsolvable, in 1958. Their work also established the unsolvability ofthe problems next mentioned by Reidemeister: deciding whether a given finitely-presented group is free,or trivial. It may be worth mentioning that Reidemeister could have had some intimation of the comingwave of unsolvability results, because he organized the conference in Knigsberg in 1930 at which Gdelfirst announced his famous (and related) result on the incompleteness of formal systems. (Translatorsnote.)


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2.11. COMMUTATIVE GROUPS 41

group on non-free generators, or whether it follows from the relations Rk(S) thatall the Siequal the identityE.

We make a simple application of the transformation rules to the modular grouppresentation by the generators S1,S2and relations

R1 = S31 1; R2 = S

22 1.

As we have seen, the operationsSand T defined in Section 2.9 also generate themodular group. We now ask what are the defining relations in the group generatedbyT = S1S22and S= S2. For this purpose we takeTas a generator in addition to S1and S2and add

R3 = S1S12 T

1

as a third relation.With the help of this equation we now eliminate S1fromR, by first constructing

R1 = R13 R1= TS2S

21and then deriving R1as a consequence ofR

1and R3. Then we

replaceR1byR1 = S1TS2S1, and this in turn by

R1 = R13 R

1 S

11 R

13 S1 = (TS2)

2.

In this way we obtain the defining relations of the modular group in the generatorsS2 = SandTas

R1 = (TS)3 1 and R2 = S

2 1.

2.11 Commutative groups

We will use the theorem of TIETZE to characterize the commutative or abeliangroupFwith finitely many generators and relations through properties of these rela-

tions. In a commutative group with generators Si(i= 1,2.. . , n) each of the relationsRi k(S) = SiSkS

1i S

1k (1)

holds, since this says thatSi andSkcommute with each other. It follows that allpower products of theSicommute with each other. Hence each relationR(S) maybe brought into the form

R(S) = Sr11S

r22 S

rnn.

Thus we can take the system of defining relations to be in the form

Ri(S) = Sri11 S

ri22 S

rinn (i= 1,2,... m). (2)

The characteristic properties of a particular commutative group must then reside inthe relations (2), because the relations (1) are satisfied in any commutative group.

We now construct the matrix

= (ri k) (i= 1,2,..., m;k= 1,2,..., n)

and show that Fhas certain characteristic numbers that may be derived from,theso-called elementary divisors of.


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By(k)i (i =1,2,...,k m,n) we mean the collection ofk-rowed subdetermi-

nantsobtainable from by striking outmkrowsand nkcolumns. If all(s+1)i = 0

while there is a (s)i = 0 thensis called the rank of. By(k) > 0 we mean the great-

est common divisor of all the (k)i fork s. Then(k) is always divisible by(k1) ,

because all thek-rowed determinants are linear combinations of(k 1)-rowed de-terminants. We now set

d1 =(1); (k) = dk

(k1) (k= 2,3,..., s)

and calldkthe kth elementary divisor of. We claimTheorem 1.The dk= 1 and n s are the same for all relation systems forF.Theorem 2.New generators

T1, T2, . .. , Tn

may be introduced, for which the defining relations take the form

Ri(T) = Tdi

i (i= 1,2,..., s).

By leaving out the generatorsTjfor whichdj = 1 one obtains a unique normalform forF; the number of relation-free generators is ns. On the basis of Theorems1 and 2 we then have: Fis characterized by the elementary divisors of the matrixdifferent from1 and the difference n s between the number of generators and therank of.

2.12 A theorem on matrices

To prove the theorems of the last section we first define an equivalence of matriceswith respect to the following transformations. The matrix = (ri k) is called equiva-lent to the matrix = (ri k)

1. if results fromby an exchange of rows or columns,

2. if results fromwhen the elementsr1iof the first row are replaced byr1i=r1i+ a r2i, or when the elements ri1 of the first column are replaced byri1 =ri1 + a ri2 (aan arbitrary integer), while all the remaining rows or columnsremain unaltered,

3. if all elements in some row or column have their signs reversed,

4. if there is a chain of matrices1 = ,2,3, . . . ,n= in whichi+1results

fromiby one of the elementary transformations 1, 2, or 3.[Again, this ndoesnot denote the number of generators.] E.g. it is a permissible transformationto addktimes a row, or column, to any other row or column, respectively.

Theorem 1. Equivalent matrices have the same rank and the same elementarydivisors.


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2.12. A THEOREM ON MATRICES 43

This is clear for matrices convertible into each other by the transformation 1.If results from by a row transformation 2, then any of its determinants (k)iresults from (k)i whenri kis replaced byri k, hence it equals (k)i if the first row doesnot contribute any elements to (k)i . Otherwise, we expand

(k)i along the first row

and obtain

(k)i =

(k)i or

(k)i =

(k)i + a

(k)j

according as the second row appears in(k)i or not. It follows that the ranks of

satisfiess sand that the elements (k) of are divisible by(k) . But sincealsoresults from by a row transformation 2, because r1i =r1i a r

2i, it follows that

s= s and (k) =(k). Hence we have Theorem 1 for arbitrary equivalent matrices.To clarify the meaning of the dkwe now assert:Theorem 2. The matrix is equivalent to the matrix = (di k), where di k= 0if

i= k , di i= di (i= 1,2,..., s ) and di i= 0for i> s.We first prove the following Lemma 1: if|ri k| > d1for all nonzero ri k, then there

is a matrix equivalent tothat contains a nonzero ri1k1 smaller than all|ri k|.Namely, letri1k1be a term ofof smallest absolute value:

|ri k| |ri1k1 |.

Now suppose there is either an element ri1k2 of thei1th row, or an elementri2k1 ofthe k1th column, which is nonzero and not divisible byri1k1 . Then by subtraction ofa suitable multiple, either of the k1th column from the k2th column, or of the i1throw from thei2th row, we obtain a matrix with the property claimed.

If, on the other hand, all theri1kand ri k1are divisible byri1k1 , then one can con-struct, by elementary transformations, an equivalent matrix in which all elementsri1k= r

i k1

= 0, except forri1k1 = ri1k1 . Then if there is an ri kwith

|ri k| < |ri1k1 |

there is nothing more to prove. If all

|ri k| |ri1k1 |

then certainly not allri kare divisible byri1k1 , otherwise

d1 = d1 < |ri1k1 |.

Ifri2k2 is not divisible byri1k1 then I construct by adding thek2th column to the

k1th column, whenceri k1 = r

i k2

(i= i1); ri1k1 = ri1k1 ,

and the second case is reduced to the first.From this we get Lemma 2: For each matrix there is an equivalent = (ri k)

with

r11 = d1; r1i= r

i1 = 0 (i= 1).

Firstly, by Lemma 1 there is an equivalent matrix containing an element equal tod1. I can bring this element into the first row and first column, and then make all


51/173


other elements of the first row and column zero by subtraction of suitable multiplesof the first row and column. This is the desired matrix.

By we mean the matrix that results from by striking out the first row andfirst column. The rank of equalss 1, essentially because anyl-rowed nonzerodeterminant from can be used to construct an (l + 1)-rowed determinant of

that is likewise nonzero.Now Theorem 2 comes about as follows:Let d2 = F22 be the greatest common divisor of the nonzeror

i k. Then d1 is a

divisor ofd2, sinced1is a divisor of allri k, and by Lemma 2 there is a matrix

=

(r

i k) equivalent towith

r

11 = d2; r

1i= r

k1 = 0 (i,k= 1)

But thenitself is equivalent to the matrix = (ri k)with

r

11 = d1; r

22 = d

22r1i= r

k1 = 0 (i, k= 1); r

2i= r

k2 = 0 (i, k= 2)

ri+2,k+2 = ri k (i,k> 1).

By iteration of this process we find thatis equivalent to a matrix = (Fi k)with

Fi k= 0 (i= k); ri i> 0 (i= 1,2,..., s), ri i= 0 (i> s)

andri iis a divisor ofFi+1,i+1.But theri iare the elementary divisors of and hence also of , because allk-

rowed subdeterminants fromthat are nonzero have a value

ri1 ,i1 ri2,i2 rik,ik

where all the il(l= 1,2,..., k) aredifferent. Sucha product is divisible byr11r22 rkk ,so

(k)

= r11r22 rkk

and hencerk k= dk= dk.

One more remark: ifdi= 1 then also di+1= 1, becausediis a divisor ofdi+1.

2.13 Characterization of commutative groups

We now return to the commutative group Fand see how the matrix = (ri k)ofthe exponents ri kin the defining relations (2) of Section 2.11 are altered when we

transform the generators and defining relations as in Section 2.10.IfRis a consequence relation of the Rithen, by means of the relations (1) of

Section 2.11 for exchange of factors, Rmay be written on the one hand as a powerproduct R

p11 R

p22 R

pmm and on the other hand it may be brought into the form

Sr11S

r22 S

rnn.


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2.13. CHARACTERIZATION OF COMMUTATIVE GROUPS 45

We therefore must have

ri=

mk=1 p

krk i.

Thus if we extend the defining relations Riby addition ofRm+1 = Rand constructthe matrix of coefficients = (ri k) for the new system, then

ri k= ri k; rm+1,k=m

i=1

piri k; (i= 1,2,..., m; k= 1,2,..., n).

One can now replace by an equivalent matrix that contains only zeros inthe (m+ 1)th row by successively subtractingpitimes theith row from the last row.Sinceresults from by omitting the last row, the elementary divisors and rank ofand, and hence also ofand, are identical.

Now letTbe any power product

T= Sq11S

q22 S

qnn .

TakeTas a new generator and

Sq11 S

q22 S

qnnT

1

as a new relation. The new coefficient matrix is then = (ri k), where ri k = ri k

(i =1,2,...,m; k =1,2,..., n); ri,n+1 = 0 (i =m+ 1); rm+1,k = qk (k =n+ 1); andrm+1,n+1= 1. We can convert into a matrix by successively addingqktimesthe last column to thekth column. In all elements of the (m+ 1)th row and the(n+ 1)th column apart from rm+1,n+1 = 1 are zero. Nowd

1 is certainly equal to

1, becauserm+1,n+1 = 1. Further,di =di1, because all i-rowed nonzero deter-

minants from are either determinants from or else they contain the element

rm+1,n+1and hence are equal to an(i 1)-rowed determinant from . Conversely,from each (i 1)-rowed determinant ofwe can construct an i-rowed determinantof with the same absolute value by taking suitable elements from the (m+ 1)throw and the (n+ 1)th column. Since all (k)i are divisible by

(k1) we have

(k) =(k1).

Therefore the elementary divisors diof are equal todi1for i>1, and d1 = 1.

From the connection between the determinants ofand it also follows that theranks of is equal tos+ 1. Consequently, the ranks of is alsos+ 1. Theorem1 of Section 2.11 then follows.

To prove Theorem 2 in Section 2.11 we show that the transformations definedin Section2.12may be accomplished for the matrix of exponents ri kby alteration

of the generators and defining relations. Transformation 1 may be accomplishedby changing the numbering of generators and relations, and transformation 3 bychanging to the inverse of a generator or relation. We accomplish transformation 2by first taking the consequence relationR1Rk2 ,

Sr11+k r211 S

r12+kr221 S

r1n+kr2n1 = R

1.


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But thenR1 = R1Rk2 is a consequence relation ofR

1,R2, . . . , Rmand hence may be

omitted. We accomplish the column transformation 2 by taking the new generator

S2and relation Rm+1 = S12 Sk1 S2. ThenS2 = Sk1 S2 and if we now replaceS2 in allrelations bySk1 S

2, usingRm+1, then

Ri= Sri1+k ri21 S

ri22 S

rinn .

TheRiare consequence relations of the Ri (i = 1,2,..., m+ 1). But conversely, theRiare also consequence relations of theRi (i= 1,2,..., m) andRm+1, since indeedRi =R

iR

ri2m+1. Consequently, the R

i andRm+1 form a system of defining relations,

and hence so do the Rialone, whenS2and Rm+1are both omitted. Theorem 2 inSection 2.11 now follows from Theorem 2 in Section 2.12.

The word problemmay be simply solved for a commutative group in the nor-mal form given by Theorem 2 of Section 2.11. All representations of the identity arecomprised by

Rk11 Rk22 R

kss = T

k1d11 T

k2d22 T

ksdss .

Thediare zero fori> s. If

Tn1

1 Tn2

2 Tnn

n and Tn1

1 Tn2

2 Tn

nn

are two words in theT, then they are the same element if and only if

ni ni (moddi).

If alldi= 0 the group is called a free commutativeor a freeABELIANgroup. FreeABELIANgroups are characterised by the number of their generators.

2.14 Commutative groups with operators

Using the coefficients defined in Section 1.13 for a commutative group with operatorx, the concepts of generator, relation, and defining relation may be extendedas follows.18 The elements

S1, S2, . .. , Sn (1)

are called the generators of the commutative groupFxwith operator when each ele-

ment ofFmay be written as a power product

n

i=1S

fi(x)i . (2)

Such a product is called a relation when it is equal to the identity element of thegroup. The relations

R1, R2, . . ., Rm

18J. W. ALEXANDER,Trans. Amer. Math. Soc. 30(1928), 275.


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2.14. COMMUTATIVE GROUPS WITH OPERATORS 47

are called defining relations ofFx in the generators S if each relation R(S) may be

derived, by rearrangement of terms, from a product

ni=1

Rgi(x)

i . (3)

Conversely, given any system of generators

S1, S2, . .. , Sn

and a system of relations

Ri(S) = Sri1(x)1 S

ri2(x)2 S

rin (x)n (i= 1,2,..., m)

there is always a commutative group with operator defined by this system. To provethis we introduce new symbols

Sxk

i = Si,k (i= 1,2,..., n;k= 0, 1, 2,...)

and setS

anxn+an+1xn+1++an+mxn+m

i = Sani,nS

an+1i,n+1 S

an+mi,n+m.

The relationsRl(Si) may be transcribed as relations Ri(Si,k) in theSi,k. We includeall relations (Rl)x

p= Rl,p(Si,k) (p= 0, 1, 2,. . .) expressed in the Si,k. Then there is

a commutative group Fgenerated by the Si,kand defined by the relations

Rxp

l = Rl,p(Si,k).

In this group the mapping defined on power productsFof the Si,kby

A(Si,k) = Si,k+1

A(F1F2) = A(F1)A(F2)

is an automorphism, because it sends each power product Rl,k to the power productRl,k+1 and hence each relation goes to another relation. Thus ifF1and F2are twodifferent power products in the Si,kwhich denote the same element ofF, so

F1 = F2R

(i.e., F1is convertible to F2Rby rearranging andapplying therelationSai,kSbi,k= S

a+bi,k ),

thenA(F1) = A(F2)A(R),

and hence also

A(F1) A(F2).Likewise, one concludes from

F1F2 F12

thatA(F1)A(F2) A(F12).


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Furthermore, the mapping Ais invertible, and hence it is an automorphism ofF.If we now introduce the exponentxinto the group by setting

A(F) = Fx

then one sees thatSi,k= S

xk

i,0 = Sxk

i

and that we have a system of generators Siand a system of defining relations Rl(Sk)for the group when the exponents f(x) are admitted.

2.15 Characterization of groups with operators

As for ordinary groups, one can ask how the various ways of defining a group withoperators by generators and relations are connected to each other. It is clear that

one can add any consequence to the defining relations, or omit any relation Rmwhen it is a consequence of the others. Likewise, it is permissible to introduce anew generatorSn+1defined as a power product of theS1,S2, . . . ,Snwith the help ofa new relation, or to eliminate a generator Snthat may be expressed in terms of theothers. One can then prove, by considerations quite similar to those in Section 2.10,that any two systems of generators and defining relations may be converted to eachother by such steps.

As a result, the properties of the defining relations characteristic of the group Fwith operatorxcan be given purely formally as matrix properties. If

Ri(Sk) =

ni=1

Srik (x)k

are the defining relations ofFxand = (ri k(x))

is the matrix of exponents ri k(x), and if

Rm+1 =

mk=1

Rpk(x)

k =

ni=1

Srm+1,i(x)

i

is a consequence relation, then

rm+1,i=

mk=1

pkrk i.

If we addRm+1to the others as a defining relation, then the exponent matrix of thenew system will be denoted by = (ri k). The passage fromto

, as well as from to, will be called a type I rearrangement of matrices. If

ni=1

Srm+1,ii


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2.15. CHARACTERIZATION OF GROUPS WITH OPERATORS 49

is any power product, Sn+1is a new generator and rm+1,n+1 = 1, and if we add Sn+1as a new generator and

Rm+1 =n+1i=1

Srm+1,ii

as a new relation, then the matrix corresponding to the new system will be denotedby. The passage fromto and conversely will be called a type II matrix rear-rangement.

The properties of exponent matrices invariant under rearrangements of the first

and second kind characterize the groupF.

It now remains to show that the elementary divisors ofmay also be defined inthis case, that the elementary divisors = xn are the same for all presentations ofFx,but that they do notchacterize Fx.

For this purpose we introduce the concept of divisibility and greatest commondivisor for L-polynomials with integral coefficients. We call f(x)divisible byg(x)

when there is a polynomialh(x) for which

f(x) = g(x)h(x).

By the greatest common divisor d(x) of the polynomials fi(x) (i= 1,2,..., r),

d(x) = (f1(x),f2(x), . . . ,fr(x))

we mean a polynomial which is a divisor of all the fi(x)and divisible by all theircommon divisors. We show that greatest common divisors always exist, and ifd1(x)andd2(x) are both greatest common divisors of the fi(x) then

d2(x) = xnd1(x).

Further: ifd(x) is a greatest common divisor of the fi(x) (i= 1,2,..., r), and iffr+1(x) =

ni(x)fi(x)

is a linear combination of the fi(x), thend(x) is likewise the greatest common divi-sor of

fi(x) (i= 1,2,..., r+ 1).

Elementa

Introduction to Combinatorial Topology-Reidemeister (1932)

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