Presentations of Groups 0521208297, 9780521208291

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London Mathematical Society Lecture Notes Series 22

Presentations of groups D.l.JOHNSON

CAMBRIDGE UNIVERSITY PRESS CAMBRIDGE

LONDON· NEW YORK· MELBOURNE

Published by the Syndics of the Cambridge University Press The Pitt Building, Trumpington Street, Cambridge CB2 1RP Bentley House, 200 Euston Road, London NW1 2DB 32 East 57th Street, New York, NY 10022, USA 296 Beaconsfield Parade, Middle Park, Melbourne 3206, Australia

©

Cambridge University Press 1976

Library of Congress Catalogue Card Number: 74-31803 ISBN: 0 521 20829 7 Printed in Great Britain at the University Printing House, Cambridge (Euan Phillips, University Printer)

To Veronica

Contents

page Preface Introduction

v 1

l.

Free groups

2.

Free presentations of groups

19

3.

Abelian groups

29

4.

More examples

36

6

5.

Tietze transformations and van Kampen diagrams

52

6.

Coset enumeration

63

7.

Presentations of subgroups

75

8.

Finite or infinite?

86

9.

Elements of homological algebra

97

10.

Cohomology of groups

111

11.'

Extensions of groups

124

12.

mod p cohomology

137

13.

Presentations of group extensions

148

14.

Minimal presentations of direct products

161

15.

Minimal presentations of wreath products

170

16.

Cyclically presented groups

182

Guide to the literature

194

References

196

Index of notation

198

Index

201

iii

Preface

These notes form the text of a course of lectures first given in Autumn 1972 and Spring 1973 to first year postgraduates at the University of Nottingham.

The first six chapters comprised an optional course for

final-year undergraduates, who have given it a more favourable reception than is usually accorded to advanced courses in group theory. Assuming only a knowledge of elementary linear algebra and a few basic facts in group theory, . it is our intention to foster dexterity in the investigation of the properties of groups specified by means of generators and relations, to make the student acquainted with a large number of examples of individual groups, and to lay a firm foundation for study in various branches of group theory of current research interest. The identification of a group given by generators and relations is not unlike a qualitative analysis in inorganic chemistry; there are various tests to be applied, whereby the set of possibilities may gradually be whittled down, ideally to a singleton.

The main difference is that in our

case there are an infinite number of possible answers. In the words of Magnus, Karrass and Solitar, there is no procedure for solving the word problem which will work for every presentation; in a sense then, every solution of the word problem for a class of presentations is a triumph over nature. Finally, my gratitude is due to a host of people whose names, too numerous to mention here, may be found scattered through the pages of the text.

Special thanks go to Professor Sandy Green for first introducing

me to research mathematics, to Dr Alan Camina for his careful criticism of the typescript, to Dr. E. F. Robertson for his help in correcting the proofs, and to my wife for her patient forbearance during the traumatic transformation of my original grubby lecture notes into this volume. D. L. Johnson

v

Introduction

On first being shown the definition of a group, a natural reaction is to ask for some examples.

Of the many ways of exhibiting or presen-

ting a group the simplest and most direct is to give its multiplication table.

gi~es

Thus the table w

G'

{3

y

6

E

w

w

G'

{3

y

6

E

a

a

{3

w

6

E

y

{:5

{3

w

a

E

y

0

y

y

E

6

w

J3

a

6

6

y

E

a

w

{3

E

E

6

y

{3

a

w

a complete description of the symmetric group S 3 •

Two advantages

of this method of presentation are its clarity and the fact that no technical knowledge is needed to appreciate it.

Two disadvantages are that it takes

up a lot of space and that the associative law must either be taken on trust or verified by a tedious calculation. Both these disadvantages (and one of the advantages) are absent if this group is specified by means of a concrete representation, for example, as the group of all permutations on a set of three elements, or as the set

of 2 x 2 matrices over C, with the usual matrix multiplication and w

= exp(271i/3).

A third concrete realization of this group is provided

by the group of symmetries of an equilateral triangle in Euclidean 3space.

1

Then again, a group is determined up to isomorphism by its graph and with a bit of luck, this can be drawn in Euclidean 2-space. a set of generators of a group G, then the graph to X is constructed as follows.

If X is

r of G with respect

There is one vertex gv for each element

g of G, and there is a directed edge bearing the label x from gv to hv if and only if gx = h, x

and valency 2lx 1.

E

r is thus a regular graph of order !G I

X.

With respect to generators x, y of orders 3, 2

respectively, the graph of S

3

looks like this:

X

A slightly more bizarre method of exhibiting this same group is to remark that it contains a subset { ~. TJ } such that the elements of the group are as follows: 2

e, ~. ~ '

T),

~T), 77~.

Since this group plainly has order 6 and is non-abelian, it must be S . 3

Incidentally, it would be interesting to know precisely which groups can be specified in this way. Perhaps the most elegant method of presenting this group is by means of generators and relators;

(1)

2

In this expression, the symbols to the left of the vertical line are the generators, and to the right stand the relators (or when convenient, the relations - obtained by equating each of the relators to the identity).

The

resulting group is then the set of all finite products of the generators and their inverses, two such products being identified when and only when their equality is a consequence of the relations (a rigorous definition will be given later).

Note that we can be even more economical, for

the expression

(x, Ylx 3

= (xy) 2 , /=e)

yields the same group.

(2)

For the relations in (2) obviously follow from

those of (1), while on the other hand, using the fact that y X

3

'~ X =¢·

= (xy)

= y

-2

2

2 "'*X = 2 -1

xy = y

yxy "'* X

2

2 X = -1

y = (y

y

-1

2

= e,

xy

2

xy) =

X

4

x 3 = e,

so' that the relations of (1) are consequences of those in (2).

One advan-

tage of presenting a group in this way is that the associative law holds automatically, being built into the process whereby such presentations are defined. It is always possible to derive a presentation of a group whose

multiplication table is known. m: G

X

Given the group multiplication

a-a,

we take X= G, R = lx y ((x, y)mf 1 ix, y

E

G],

whereupon G

= (XjR).

But the converse problem is not so simple; its solution depends on that of the so-called 'word problem' for the presentation in question, and this cannot always be decided. In fact, there even exists a presentation for

3

which it is impossible to decide whether the word problem is soluble or not.

Such mind- bending phenomena are fortunately beyond the scope of

these notes - we restrict ourselves to the study of relatively non-pathological presentations. Even with this restriction, the derivation of the properties of a group from a given presentation is by no means an easy task.

For

instance, it is not obvious that Conway's group (n, {:;, y,

o,

c/af3 = y, {3y=

o,

yo= E, liE= n, t:n = {3)

is in fact cyclic of order 11, although simple manual manipulation of the relations suffices to prove this. On the other hand, the group (a, {3, y,

o,

c/nf3y=li, f;yli=E, yliE=n, liEn={3, Ea{3=y)

has been identified as Z

22

using coset enumeration on a high-speed

computing machine, but some 200 cosets are required and no short manual proof has yet been found. Before developing the powerful techniques needed to handle more complicated situations, it is necessary to say precisely what we mean by a presentation, and this leads us into the theory of free groups, the elementary properties of which are described in the first chapter.

It is

then a simple matter to define the concept represented by the symbols (X jR) (Chapter 2), and the next two chapters give a variety of important examples.

The geometrical process (Chapter 5) for depicting the deriva-

tion of a new relation from a given set of relations is intended as an aid to comprehension and a guide to intuition, although it has recently proved to be of theoretical importance in the theory of decision problems mentioned above.

The subject of the next chapter is probably the most impor-

tant general method of dealing with group presentations, and is currently in frequent use on high-speed computing machines.

Chapter 7 is adequate-

ly described by its titlt, while Chapter 8 is devoted to the description of a number of simple criteria for deciding whether or not a given presentation yields a finite group.

Most of these have already appeared in the

text, while the justification of others is too technical to be included.

4

The first half of the notes thus represents a fairly relaxed treatment of the basic ideas and methods in the theory of group presentations, and there are at this point a number of avenues open to us.

For example,

we could proceed to delve more deeply into the combinatorial theory suggested by Chapter 1, or investigate the geometrical problems arising in Chapter 8, or give an account of the recent work of Stallings and others on ends of groups.

The way is paved for an excursion into these and many

other aspects of the theory of group presentations.

However, the most

important single tool in this subject is provided by the cohomology theory of groups, as is borne out by the fact that the most impressive of recent results in this field is proved by a cohomological method.

This is the

famous theorem of Golod-Safarevic and has to do with the minimal number of relations needed to define a finite p-group.

We therefore bend our

steps towards the theory of minimal presentations - a region where there is still room for elegant proofs and scope for human ingenuity. So we plough through the abstract world of homological algebra to the cohomology of groups, and thence to the theory of group-extensions, a subject of paramount importance in its own right.

This, together with

the subsequent discussion of localized cohomology prepares the way for the striking theorem of Golod-Safarevic, and for Raquette's equally striking proof.

We pass on to the theory of minimal presentations, and

give a couple of simple examples to show how little is known about this rather intractible subject.

We close with a light- hearted look at an enter-

taining topic which has recently aroused the interest of a number of mathematicians, and solve a handful of problems by methods which are as remarkable for their diversity as they are for their simple-mindedness.

5

1· Free groups

The fundamental notion used in defining presentations of groups is that of a free group.

As the definition shows, the idea of freeness is

applicable in algebraic situations other than group theory. Definition 1.

A group F is said to be free on a subset S

1 in G and /y/

>

1 in H, then lxy/ in

This provides the basis for a number of tests for in-

finite order, but that described above is the simplest and the most useful. Example 3.

The two simplest examples of non-trivial free prod-

ucts are the groups Z

2

* Z 2 = (x,

ylx 2 , y 2 ),

= (x,

y/x 2 , y 3 ),

Z • Z 2

3

89

and in each of these the element xy has infinite order.

The first of these

groups is just the infinite dihedral group

, while the second is the modular group PSL(2, Z) - the group of 2 x 2 matrices over Z of determinant 1, factored out by the normal subgroup {±12 } .

Our next criterion again depends on material outside the scope of the notes, and so we cannot prove that the test works.

It is based on

the Golod-Safarevic Theorem (Chapter 14) and a result of Wamsley on the multiplicator.

The criterion asserts that if G is presented with an

equal number of generators and relations, and if G/G' has at least four invariant factors, then G is infinite.

Example 4.

We consider the Fibonacci group

F(4, 5) =(a, b, c, d, e labcd=e, bcde=a, cdea=b, deab=c, eabc=d; and note that the same method works for F(n, n+ 1) whenever n

2:

4.

Using a Tietze transformation to eliminate the generator e, we obtain the presentation (a, b, c, dlbcdabcd=a, cdabcda=b, dabcdab=c, abcdabc=d), and since all the exponent sums occurring here are even, it follows that GIG' has four invariant factors.

Whence G is an infinite group.

The fifth and final criterion is to exhibit G, or possibly a factor group of it, as a concrete group which is palpably infinite.

This involves

letting G act in some definite way on a given set - with or without structure.

There are many ways of doing this, and we shall limit ourselves to

describing the three most common.

The first is to realize G as an in-

finite permutation group, usually on the set Z of integers.

This was

done for the group

I

F(2, 6) =(a, b, c, d, e, f ab=c, bc=d, cd=e, de=f, ef=a, fa=b) in Exercise 2. 7, where the action is given by

90

n(ax) = n + 1, n(bX:) = -n . We see that Imx

~

D ""' the infinite dihedral group. It may also be pos-

sible to realize G as an infinite matrix group, as in the following example. Example 5.

We set up a homomorphism (Theorem 2. 3) from

F(2, 6) to GL(2, Z) by letting the generators a and b be mapped to

(~ ~) respectively. (1

2

' The images of c, d, e, f are then

0)

( 1

0)

(1

0)

(1

-1 '

-2

1 '

4

-1 '

2

and there results a homomorphism: F{2, 6) - GL(?., Z) because 0)

1 '

(1

0)(1

2 -1 2

0)

1

= (1

0)

0 -1 •

Since the image of a has infinite order, so does a, and we have another proof that F(2, 6) is infinite. Finally, we give three examples of some historical importance to show how a group can be made to act automorphically on a geometrical configuration.

Unlike the two examples involving F(2, 6), the action is

faithful in all three cases, and the configuration used is in fact the graph of the group in each case. Example 6.

Consider the von Dyck group D{4, 4, 2), given by:

The following diagram indicates how this group can be made to act on the vertices of the square grid in Euclidean 2-space.

Note that the square

ACEF repeats indefinitely in all four directions.

The set V of vertices

may be thought of as the set of Gaussian integers on which G is acting as a group of permutations. Now each vertex v has the property that there is precisely one edge labelled x having initial point v. the action of x on V.

This gives

Thus x induces a mapping

91

X

X

y y

y

y

y y

y

y

y

~

X

X X

X

X

A

X

X

F

X

y

y

y

y

y

y

y

y B I

X

y y

X

X

X

X

X

X

X

X

X ~

c

D

x':

v-v,

Cx'

= B,

y

E

y

with

for example.

Dx'

= C,

The same is true for y, so we have a map y' : V - V,

sending A to B and D to E, for example. Since each v

E

V is the terminal point of exactly one edge labelled

x, it follows that x' is a bijection, and similarly for y.

Now for each

relator r defining G and each vertex v in the graph, there is a face of the graph which is incident on v and whose boundary is r. It follows from this that the mapping xl-+x',

92

y~-+y'

extends to a homomorphism from G into the symmetric group on the set V, by Theorem 2. 3.

Since x' 2 y' 2 has infinite order, it follows that G

is an infinite group.

The problem of proving that this action is faithful

is consigned to the exercises. Example 7.

The group D(3, 3, 3) is given by

and can be made to act on an infinite lattice in Euclidean 2-space, as indicated by the following diagram.

Here, the chevron ABCDEF repeats indefinitely in all six directions. By the same arguments used in Example 6, we obtain an action of G on the vertices of the graph, and the order of y- 1 x is seen to be infinite, whence G is an infinite group.

93

Example 8.

We apply the arguments of the previous two examples

to D(3, 6, 2), given by

This time, the lattice is as follows:

The section bounded by heavy lines is understood to repeat indefinitely as above until the plane is covered. 2 -1

y x

94

is clearly infinite, so G is an infinite group.

The alert reader will no doubt have observed that we

Remark.

have now dealt with all von Dyck groups D(l, m, n) for which

~+~+~2::1 l

m

n

'

the three cases of equality in this chapter, and all others in the foregoing. Thus, D(2, 2, n) is the dihedral group D of order 2n, while the groups n D(2, 3, n) for n = 3, 4, 5 are respectively the tetrahedral group A , 4

the octahedral group S

and the icosahedral group A . The geometrical

4

5

significance of these groups is no accident, and this has been exploited by Coxeter to give an elegant proof of an old result of Miller to the effect that the group D(l, m, n) is finite if and only if

When this sum is less than 1, the proof involves the action of the group on a configuration in the hyperbolic plane, rather than the Euclidean plane. Again, a more detailed treatment is beyond our scope and already exists elsewhere. Exercises. G

= (x,

Show that x 7y 7

1.

Consider the group

y Ix 2 yxy 3 , y 2 xyx 3) .

=e

in G.

Use the fact that

IG : ( x) I = B

(Exercise

6. 3) to give (yet another) proof that G is finite. 2.

Show that the group

C(m, n)

= (x,

is finite for all m, n (m, n)

yixm E

N.

= yn = [x,

y])

Find the order of C(m, n) in the case when

= 1. 3.

Prove that the Fibonacci group F(n, n+1) is infinite whenever

n 2:: 4 (see Example 4).

4.

Prove that the representations of the Euclidean von Dyck groups

given in Examples 6, 7 and 8 are faithful.

95

5.

Identify the group whose graph is as follows:

X

As in Examples 6, 7 and 8, the section bounded by heavy lines is understood to repeat until the plane is covered.

96

9 · Elements of homological algebra

Although entirely algebraic in content, the subject of homological algebra has its motivation in algebraic topology, out of which it grew some thirty years ago.

Hence parts of the nomenclature, as well as many

of the more powerful and lucrative methods, are topological.

There is no

doubt that the discovery of homological algebra was a milestone in the history of pure mathematics; over the last thirty years, homological ideas have pervaded all branches of the subject, from foundations to algebraic geometry and class field theory.

Not least among the domains of

its application is the theory of groups, and we plan to exploit one aspect of this connection in the course of the next few chapters. We will be dealing with rings and modules, and will assume throughout that our rings have identities and that our modules are right, unital modules. Our first result has to do with induced homomorphisms and is the 'module-theoretic analogue of Lemma 2. 1. Theorem 1.

Let X and Z be modules over a ring R anci__l~

{3 : X -+ Z be an R- ~omomorphism.

If A is a submodule of X contained in Ker {3, then f3 in-

(i) ~ces

an R-!J.omomorphism {3' : X/A -+ Z } A+x...,. x{3 ·

(ii) thel!__!~e~e

If a : X -+ Y is an onto R- homomorphism with Ker a::= Ker (3, ----

is an

Proof.

R-~~~omorphism

y : Y-+ Z such that ay

= {3.

To prove (i), observe that the hypothesis A -c::: Ker {3 im-

plies that the value of {3' is independent of the choice of coset representative. A

The R-linearity of {3' is easy to check.

= Ker

For (ii), apply (i) with

a to get the R-linear map

97

(3' : X/Ker a-+ Z ] Ker a + x ._. x(J •

Composing this with the natural isomorphism Y

= Im

a

2=

X/Ker a,

we obtain the required y. The theory of categories provides a useful language and context for the development of homological ideas.

We shall not give a formal

(£

definition of a category, but merely remark that a category of a class

Il£ I,

consists

the objects of [, and a set Hom(A, B) - the morphisms

from A to B - for each pair A, B

E

I(£ 1.

We are also provided with a

mapping Hom(A, B) x Hom(B, C) ._. Hom(A, C)} (a, (3)

1-+

a(3

'

called composition of morphisms, for all A, B, C position is associative. phism l A

E

Finally, for each A

E

I:

D.

(13)

Applying * to (12), the functorial properties of * to-

gether with (13) yield that

a~= (?nTJn + TJn+1Yn+1)*

= (on7Jn)*

+ (7Jn+lyn+1)*

-- 7]*2* n 'n + y*n+1 1]*n+l' which plainly maps Ker

y~+ 1

on the cohomology groups. 0 =a*= (i\- f..Ll*

into Im

a~,

and so induces the zero map

Thus

= i\*-

f..L*,

establishing our claim. The end is now in sight.

Let

i\ : p -+ Q,. IJ. : Q-+ p

be chain maps of degree zero induced by l A'

Then

i\f..L : P -+ P, f..Li\ : Q -+ Q

are chain maps of degree zero ind-.1ced by 1A'

But so are

109

1p : P-+ P,

1Q : Q-+ Q.

The claim proved above now guarantees that

l P*

= 1p* = (.\,.. )* = ,.. *1..* 11

11

,

which is the identity on the cohomology groups defined by P. >.. *!l*

is the identity on the cohomology groups defined by Q.

Similarly, Hence,

A.* is an isomorphism for all n. n

Note.

This theorem entitles us to write the group Ker

il~+ 1 ;Im

a;

unambiguously as Ext; (A, B). Exercises.

1.

Prove the generalized associative law for the

composition of morphisms in an arbitrary category. 2.

Let A be a submodule of an arbitrary R-module B.

Prove

that A has a complement in B if and only if there is an R-homomorphism from B to A fixing A pointwise.

Use this to complete the proof

of Theorem 2. 3. Ho~(X,

For any R-module X, define a covariant functor ) : \£R-+ [Z' in analogy with Ho~( , X), and prove that it

too is left exact. 4.

Show that a ring R regarded as a module over itself is pro-

5.

Let A and B be R-modules.

jective. Prove that A ffi B is pro-

jective if and only if A and B are both projective. 6.

For any R-module A, prove that

Ho~(R,

A) is isomorphic

to the additive group of A. 7.

Let P and A be R-modules with P projective.

Prove that

the cohomology groups of P with coefficients in A are trivial in all positive dimensions.

Identify Ex~ (P, A).

denote the cyclic group of order m g

is a free resolution of k

over k(G x H). Proof.

That the R

are k(G x H)-free and that the TJ

n

n

are

k(G x H)-linear is already clear. We next check that the composite of any two adjacent TJ 's is zero.

First let p 18> q

E

P l 18> Qn-l , with 2 :::: l :::: n- 2.

Then

=

as required.

o,

The calculation is even simpler when l

assumes its

extreme values, and we omit it. Now we come to the crux of the whole business - showing that the sequence R is in fact exact.

Now TJ

-

0

is certainly onto, and for the

rest, it is sufficient to prove that Im T)n+ 1 and Ker T)n have the same dimension over k, since we already know they are comparable.

To this

end, write Im an =I n- 1' Im E n and let C , D n

P

n

n

= J n- 1'

n

~

1,

be k-spaces such that

= I n El1 C n ,

Q n

= J n El1 Dn ,

n ~ 0,

as vector spaces over k. We investigate the action of TJ

n

(n

~

1) on

Pl ® Qn-l by examining its effect on each of the four direct factors

1z 18> Jn-l' Cl 18> Jn-l' 1z 18> Dn-l' Cl 18> Dn-l .

Now for 0 < l




Q

A-+0

Then X EEl Q

Proof. 0

-+

~

Y EEl P.

We have a commutative diagram

a

f3

X -+ P

-+

t11 e

A -+ 0

ts JIA 1/>

0-+Y-+Q-+A-+0,

170

where

~

exists because P is projective, and

is induced by

7)

~.

We

define mappings L

7T

X-+PIBY x

f-+

(xa,

PIBY-+Q

}

X7))

'

(p, y) 1-+

p~

- y()

which are easily seen to be R-homomorphisms.

We claim that the

sequence 1T

0-+X-+ PIBY-+ Q-+0 is exact.

(1)

The proof is by standard diagram-chasing and we leave it as

an exercise.

Since Q is projective, (1) splits, and the result follows

from Theorem 9. 2. Now let R =kG, with G a finite p-group, and define two kGmodules A, B to be equivalent (write A - B) if there exist free kGmodules P, Q of finite rank such that A IB P

~BIB

Q.

It is easy to check that this is an equivalence relation, the transitivity

being a consequence of such elementary facts as the associative law for direct sums.

Now suppose we have free presentations

0 -+ X -+ P

-+

A -+ 0,

0 .... Y -+ Q -+ B .... 0

with P and Q finitely generated, where A and B are equivalent kGmodules. It is an easy consequence of Schanuel's Lemma that X and Y are also equivalent, and a simple application of induction yields the following result. Let

Theorem 2.

0

0 p

~ be

-+P -+Q

n n

n -+: p

£

.... . . . n-1

all n

1

n

....

Q n- 1 ....... -+Q 1

~Cl_ fre_~__!_esolutiol)~ of

ra$~!!~n_for

-+P

~

c

0

1

....

p 0

1

....

Qo

....0

k-+0

£

....0

k-+0

k over kG, with all P's and Q's of

finit~

1,

171

Im 2 - Im £ •

n

n

Theorem 3.

Proof.

Then

Let P, Q be free - of ranks p, q say - such that ~

A@ P

Let A and B be equivalent kG-modules.

B EB Q.

Then recalling that d(X) is just the dimension of X/XU (Theorem 12. S(iii)), we have d(A) + p

= d(A) + d(P) = d(A EB P) = d(B ® Q) = d(B) + d(Q) = d(B) + q,

and

di~A + pIG I

= di~A + dimkP = dimk(A ® P) = di~(B EB Q) = dimkB + dimkQ = dimkB + qiGI.

IG I

Multiplying the first of these by Theorem 4.

Let

an p

....

-+P

n£ -+Q

Q

and subtracting gives the result.

n

p

n-1

-+

. ..

a1 -+P

...n Qn- 1 ....... -+Q

1

1

....

E

a

....0

p 0

1

-+ Qo

£

....0

k-+0 k-o

be kG-free resolutions of k with P , Q of ranks p q respectively, --- n n n' n n ~ D. Suppose that p

_q n

n

= {o,, 0 :s n < m, c / 0, n = m.

Then there exists a kG-free resolution of k with the free terms of ranks

Proof. Im

172

We concentrate on the modules

am = Ker am- 1 ,

Im

E

m

= Ker

E

m- 1

(interpreting Ker A

m

, B

m

E

-1

= Ker o-1 = k

respectively.

if necessary) - call these modules

Now our hypothesis on the p's and q's implies

that dim. A

K m

= dim B . K m

Furthermore, we know from Theorem 2 that A d(A

m

- B , and so m m Since there is an epimorphism

) = d(B ) by Theorem 3. m Q

m

-B

m'

we have d(A ) m

= d(B m ) ~

d(Q ) m

= qm ,

and so there is an epimorphism 7T:Q

m

-A.

m

Now by Schanuel's Lemma, Ker rr- Ker

om ,

and (2)

whence d(Ker rr) by Theorem 3.

= d(Ker om ) - c, Since there is an epimorphism

P m+l- Im cm+l

= Ker

am'

(3)

we have that d(Ker rr) = d(Ker am) - c ~ pm+l - c, so there is a free presentation (4)

p: P - Ker rr, with P of rank pm+l - c.

From (2), (3) and (4), we have that

173

dimkKer p

= di~P = (pm+ 1 -

di~Ker 11

c)IGI- (di~Ker am- ciGI)

= pm+11G I - dimkKer am = di~Ker om+ 1. Furthermore, Ker p and Ker am+ 1 are equivalent by Theorem 2, and so d(Ker p)

= d(Ker Clm+ 1 )

by Theorem 3. Hence Ker p has a free presentation T:Pm+ 2 -Kerp. Repeated application of Theorems 2 and 3 enables us to conclude that Ker p has a free resolution by means of modules of ranks p m+2' Pm+3' · · · respectively.

Sticking all these bits together, we get a free resolution 0m-1 T' p' 11 1 ... -+Pm+2-+ p-+ Qm-+ pm-1 ........ -+Po-+ k-+0, (5)

°o

where r', p',

11'

are the composites of

inclusion mappings, Remark.

T,

p,

with the appropriate

11

The resolution (5) has the desired ranks.

This theorem effectively says that we can cancel super-

fluous copies of kG from adjacent terms of a free resolution.

It is

really little more than a technical lemma and as such it is much easier to apply than it is either to state or to prove. We now turn to the construction of the wreath product GIH of two groups G and H.

We shall assume for the sake of simplicity that

both factors are finite, and our aim will be to write down a presentation for GIH in terms of presentations of the factors. Let H =

1h 1 , ... , hn } have n elements and let B

product of n copies of G - B is called the base group. extension of H by B and so has order n IGIn.

be the direct

GIH is a split

To specify this extension,

it is enough to describe the action of H on B, and this is done as follows.

174

·

Any element h

E

H gives rise to an element a

E

S

n

given by

h.h = h. , l :::;; i :::;; n. 1

If (g 1 ,

1a

gn) is a typical element of B, we define the action of h

••• ,

by setting I

I

I'

gn)h

Example.

= (g

g

-1' • • • ' 1a

na

-1 ).

If H = Z , the cyclic group of order p, then B is p

an elementary abelian p-group admitting a linear H action.

It is thus a

kH-module, and as such is isomorphic to the group-ring kH itself.

If

G is also cyclic of order p, then the resulting wreath product is iso-

morphic to the Sylow p-subgroup of S p

Z lZ 2

2

2

, of order p 1 +p.

In particular,

is just the dihedral group D . 4

We now turn to the problem of constructing a presentation for GIH in terms of given presentations

G

= (X!R),

of G and H.

H

= (Y!S)

Repeated application of Theorem 2. 4 enables us to write

down a presentation for the base group:

B=(X' ... ,X 1

n

IR'1 ... , R'n

{[X., X.]}), 1

J

where the X. are disjoint copies of X, R. is obtained from R by sub1

1

stituting corresponding elements of X., and commutators [X., X.] are 1

included for all values of i and j such that l :::;; i


'

are made

redundant by the second set of relators, and all but the first of the first set of relators are also superfluous (cf. the analysis preceding Theorem 15. 5).

The employment of suitable Tietze transformations thus leads to

the presentation t tr- 1 { k E(r,n)=(x,tlxx ... x =x , t =e). 1

1 1

1

1

Writing x in place of tx~ 1 and eliminating the generator x 1 , we obtain the following result. Theorem l. E (r, n )

The group E(r, n) has a_ presentation

= ( x, t Itxr = xtr'

tk = e) '

(1)

183

where k

~_the

order of the automo_rphism () of F(r, n).

Now since k is finite (in fact a divisor of n), F(r, n) is finite if and only if E (r, n) is finite, and what Lyndon in fact proved was that the group (2)

is infinite for n

11.

?

Note that in the passage from (1) to (2), not only has the r become a 2, but also the k has become an n. fying those pairs (r, n) for which k

=n

The problem of identi-

leads us to the study of the

abelianized Fibonacci groups A(r, n) and in general of the abelianized G (w), which we write A (w). n

n

Thus we begin with a general study of the

A (w) with a view to obtaining information about the A(r, n) and of n

= n.

throwing some light on the problem of when k We first find a formula for

lA (w) n

I

using the theory of Chapter 3.

To this end, let a. be the exponent-sum of x. in the word w, and 1

1

define a polynomial f(x)

n

= I

a.x

i=1

i-1

(3)

,

1

the polynomial associated with w.

Since the n permutants of w under

(12 ... n) comprise a set of defining relators for G (w), it follows that n

the matrix

a a a

2

l

3

a

a a

3 2

(4)

4

is a relation matrix for A (w). n

This is a circulant matrix and its deter-

m inant is known. Theorem 2.

With the notation of (3) and (4)

n

det

c = n f(w.), i=1

184

1

where w.1 ranges over the--set of complex nth roots of unity. ------~------

Proof (Belcher).

Let w be a primitive nth root of unity, and

let V = (v .. ) be the Vandermonde matrix given by 1]

v .. -1]

w

ij •

Now the (i, j) entry of the product CV is equal to

whence det(CV)

n

= det([

.

n f(wl)]v).

(5)

j=1

But the (i, k) entry of V 2 is equal to n

l:

w

(i+k)"

=

l

j=1

* n,

0,

if i + k

n,

if i + k = n, 2n,

2n,

so that V 2 is just n times a permutation matrix.

In particular, V is

non-singular, so the theorem follows from (5). In accordance with the theory of Chapter 3, we now have a formula fot the order of A (w).

n

Theorem 3. lA (w)l n

=±

If f is the polynomial ass_ociated with w, then

n fW.

(6)

~n=1

Note that the sign in (6) is chosen so as to make the right-hand side positive, and that we observe the convention usual in this context that 0 = ""· Theorem 4.

The

f~~_owing

three assertions a_re

eq~valent:

(a)

An (w) ~~inite,

(b)

f(x) !1~3 root in COTil_rnOn with xn - 1,

(c)

f(x) !~z_e_!'o:-diylf>orln the ring Z[x]/(xn- 1).

Proof.

The equivalence of (a) and (b) follows at once from (6).

Assume (b) and let w be a common root.

Let w be a primitive kth

185

root of unity and let ¢k (x) be the kth cyclotomic polynomial. Since f(x) and xn -1 are both integral polynomials, they are both divisible by ¢k(x) -

say, where p(x), q(x)

E

Z(x].

It follows that f(x)p(x) belongs to the

ideal (xn - 1), and since the degree of p(x) is less than n, (c) follows. Conversely, let (c) hold so that there are polynomials p(x), g(x)

E

Z[x]

such that f(x)p(x)

= (x n -

1)g(x),

with p(x) not divisible by xn - 1. X

n

-

=

1

Now the factorization

II ¢k (x) k/n

into irreducibles shows that for some k, ¢k (x) is not a divisor of p(x). By uniqueness of factorization, ¢k(x) is a divisor of f(x), and assertion (b) follows. Theorem 5 (Dunwoody). a unit in the ri~ Z[x]/(xn- 1). Proof.

A (w) is trivial if and only if f(x) is n

Suppose first that f(x) is a unit, and let g(x)

E

Z [x] be

such that f(x)g(x)

= 1 (mod(xn - 1)).

(7)

We can of course assume that g(x) has degree at most n - 1, and it is a consequence of Theorem 2 that any such polynomial has the property II g( ~)

E

Z.

(8)

~n=1

That A (w) is trivial now follows from (6), (7) and (8). n

converse, consider the mapping 7T :

186

F -

Z[x]/(xn- 1)

u

h(x) + (xn - 1) ,

1-+

To prove the

where h is the polynomial associated with the word u. It is clear that rr is an epimorphism of groups, and that Ker rr sist of w and its permutants under (12 ... n).

= F'.

Now let R con-

Then G (w) n

= F ;R,

and A (w) is trivial if and only if the composite mapping n

inc rr a: R - F - Z[xl/(xn- l) is onto.

But Im a is just the subgroup generated by the set Ra, namely,

by the cosets containing f(x), xf(x), ... , x

n-1

f(x).

Thus, when A (w) is trivial, we can find integers b , ... , b 1 such n o nthat n-1

= l

1

.

b.x1f(x) 1

i=O

n-1

in Z[x]/(xn- 1), whereupon the polynomial

l: i=O

.

b.x1 is the desired 1

inverse for f(x), modulo (xn- 1). Focussing our attention once more on the A(r, n), we fix r

2:

and write a n

=

IA(r, n)

1.

Letting f be the polynomial f (x)

= xr

r-1

l:

-

.

x1

(9)

,

i=O

we see from (6) that

a = n

±

n

Theorem 6. Proof.

f(~).

A(r, n) is al'Y l,

Iz In+ 1 - 1 I I qn (z) ;::: I In z +l

=

I z I _ Iz I + 1 , I In z +l

which is increasing and exceeds l eventually. If on the other hand lz I < 1, 1 -

Iz In :s Izn -

11 :s 1 + Iz In,

so that lqn(z)! tends to l in this case.

Now in the proof of Theorem 6,

we showed that f has no root of modulus 1, and so these are the only two cases that can arise in studying the limiting behaviour of

So to prove the theorem, it is sufficient to show that f has at least one root outside the unit circle, and this follows at once from the fact that the product of its roots is ±1. Theorem 9.

For n large enough, a

n

~ceeds

all its pre-

decessors. Proof.

Since the a

n

are integers, this is an immediate con-

sequence of Theorem B. We return briefly to the problem of deciding when k

en = e

190

and k

= I0 I,

= n.

Since

k is certainly a divisor of n, and the epimorphism

F(r, n)- F(r, k)

(14)

given by reducing subscripts modulo k (see Theorem 2. 3) is an isomorphism in this case.

As a consequence, an

= \:

and Theorem 9 yields

the following result. Theorem 10. (} of F(r, n).

Let k be the order _o!_!l1e natural automorphism

=n

Then k

provicle_d

tha~

n

!s surficiently_large_in

relation to r. Theorem 11.

Given a group G, there are at most finitely many

pairs (r, n) such that G Proof.

~

F(r, n).

If G/G' is infinite, the result follows from Theorem 6.

So assume that G/G' is finite - of order m say - and suppose that G = F(r, n).

By (14), G has the group F(r, 1) as a factor group and

since F(r, 1)

~ Z

r- 1

, we must have r ::::: m + l.

So there are at most

= F(r, n), and for each of these, there are with a = m, by Theorem 9. This proves the n

finitely many r such that G at most finitely many n theorem.

We end by offering a couple of conjectures. Conjecture 1 (Dunwoody).

If G (w) is trivial, then the associated

polynomial f(x) has the form ±xi for :orne i. Conjecture 2.

.

The natural automorphism

e

of F(r, n) is trivial

when r is a multiple of n and otherwise has order n. Exercises.

l.

Prove that F(n, n+ 1) ~

~

2.

Show that F(r, n)

3.

Show that the cyclic group of order 2

Z

r- 1

F(2n-1, n) for all n =:: 2.

when n is a divisor of r. s-1

- 1 appears at

least s times among the F(r, n). 4.

Prove that A(2, n) has order f - 1 - (-1)n, where f n

n

is

the nth term of the Fibonacci-type sequence 1, 3, 4, 7, 11, 18, . . . .

191

Thus show that conjecture 2 is valid when r = 2. 5.

Show that for all s

1, r(2s+1, 2) is a metacyclic group

2>

of order 4s(s + 1).

6.

Prove that every finite group is a factor group of some

7.

Prove that when r

F(r, n).

=1

(mod n), F(r, n) is a metacyclic

group of order at most n(rn - 1). 8.

Define F(r, n, k) to be the group G (w) when w is the n

1 k" word x ... x x -+

Find necessary and sufficient conditions for

r r

1

F(r, n, k) to have an infinite abelian factor group.

9.

Show that x ... x = e in F(2, n). n

10. A 1

1

Use the matrices

=

-1 ( -1

1 0

0

0

~)

A

~ 1

=(

2

1

0 0

~) .

to show that the group F(2, 1 0) is infinite. 11.

Use the criterion involving free products (see Chapter 8) to

show that the group G =(c ,c ,c ,c lc 2 =c 2 =c 2 =c 2 =e, c c c c =c c c c =c c c c) 1

1

2

3

4

1

2

3

4

1234

2413

4321

is infinite. 12.

Show that the group G

of the previous question has an

1

automorphism of order 5 mapping c 1-+ c , c 1

2

2

~-+

c , c 3

3

~-+

c , c4 4

~-+

c1 c 2c 3c 4 ,

and use Theorem 13. 1 and Tietze transformations to show that the resulting split extension G G 2 = ( c, d 1c

192

2

2

of Z 5

5

by G

= d = (cd)

5

1

has a presentation 25)

= (cd ) = e .

13.

Prove that the group G 2 of the previous question is genera-

ted by d and cd 3 c. 14. G

Show that there is a homomorphism from F(2, B) onto 3

3

mapping x 1 to d and x 2 to cd c, and thus show that F(2, B) is

an infinite group.

193

Guide to the literature

Such prerequisites as are necessary for reading these notes may be found in [11 ], (16 ], (20] - linear algebra in the former and group theory in the other two.

[20] also contains a discussion of the word problem,

while a fuller account of the theory of decision problems appears in [19]. Knot theory is dealt with in the classical text [ 6 ], and the theory of ends in Stallings' excellent monograph [21 ], which also contains a very elegant alternative introduction to combinatorial group theory.

[4] and [1 7] are

of course of general interest throughout. The elementary properties of free groups are to be found in (16 ], (20], and the former contains a proof from scratch that IRI (X IR> is finite.

2':

lxl

when

These texts also give the Basis Theorem for abelian

groups, but the proof we need in Chapter 3 is that given in [7], under the heading of the Invariant Factor Theorem for Matrices. Mennicke's groups first appeared in (18], and Tietze transformations are discussed in [1 7].

The method of coset enumeration was intro-

duced in [5], and is also discussed in [4], which also contains a full treatment of von Dyck groups, together with many attractive diagrams of the type appearing in Chapter 8.

Presentations of subgroups are dis-

cussed in (17] and in Mendelsohn's article in [15]; this also furnishes the best reference for computational problems in group theory.

The theory of

the Schur multiplicator is dealt with in [7] and [12]. Homological algebra is introduced in (20] and covered in immense generality in [2].

The best reference for the bar resolution is the original

paper [8], which also has an account of extension theory.

I don't know of

a good reference for the contents of Chapter 12, nor have I seen an explicit treatment of the theory of presentations of group extensions as described in Chapter 13.

The Golod- Safarevic Theorem appeared in [9]

and Raquette's proof is in [3].

This and many other applications of

cohomology to group theory are to be found in [10].

194

The derivation of the local cohomology of finite abelian groups does not seem to appear in the literature in quite the same form as in Chapter 14.

The significance of the class g , together with current p

knowledge of its extent, is adequately covered in

rzz].

The presentation

theory and cohomology of wreath products is lifted directly from [13]. Most of the last chapter appears in (14 ], which contains references to other notable results on Fibonacci groups - for example, the work of Campbell and Robertson on the metacyclic F(r, n) and on more general classes of cyclically-presented groups.

Exercises 10-14 of this chapter

are a summary of the work of Brunner [1 ].

195

References

l.

A. M. Brunner.

The determination of Fibonacci groups, Bull.

AustraL_]viath. j>oc., 11, no. 1 (1974), 11-14, 2.

H. Cartan and S. Eilenberg.

Homological algebra, Princeton

3.

J. W. S. Cassels and A. Frohlich (eds. ). Algebraic number

University Press, Princeton, 1956. !heory, Academic Press, London, 196 7. H. S. M. Coxeter and W. 0. J. Moser.

4.

Generators and relations

for discrete groups, 2nd ed. , Springer, Berlin, 1965. H. S. M. Coxeter and J. A. Todd.

5.

A practical method for

enumerating cosets in an abstract finite group, Proc. Edinburgh Math. Soc. (2), 5 (1936), 25-36. 6.

R. H. Crowell and R. H. Fox.

Introduction to knot theory,

Ginn, Boston, 1963. C. W. Curtis and I. Reiner.

7.

~roup~and_li:Ssociative

8.

Representation theory of finite

algebras, Interscience, New York, 1962.

S. Eilenberg and S. Maclane.

Cohomology theory in abstract

groups I, Ann. of Math. (2), 48 (194 7), 51-78. E. S. Golod and I. R. Safarevic. On the class field tower (in

9.

Russian), Izv. Akad. Nauk SSSR, 28 (1964), 261-72. 10,

K. W. Gruenberg.

Cohomological topics in group theory,

Springer, Berlin, 1970.

ll.

B. Hartley and T. 0. Hawkes.

Rings, modules and linear algebra,

Chapman Hall, London, 1971. 12.

B. Huppert.

Endliche Gruppen I, Springer, Berlin, 196 7.

13.

D. L. Johnson.

Minimal relations for certain wreath products

of groups, Can. J. Math., 22, no. 5 (1970), 1005-9. 14.

D. L. Johnson, J. W. Wamsley and D. Wright. groups, Proc, London Math. Soc. (3), 29 (1974).

196

The Fibonacci

15.

J. Leech (ed. ).

Computatiof1~_:eroi:Jlems in_~~stra_ct ~lgebra,

Pergamon, Oxford, 1970. 16.

I. D. Macdonald,

The theory of groups, Oxford University Press,

Oxford, 1968, 17.

W. Magnus, A. Karrass and D. Solitar.

Com~inatorial

group

theory, Inter science, New York, 1966. 18.

J. Mennicke.

Einige endliche Gruppen mit drei Erzeugenden

und drei Relationen, Arch. Math., 10 (1959), 409-18. 19.

A. W. Mostowski.

Decision problems in group theory, Technical

Report no. 19, University of Iowa, 1969. 20.

J. Rotman.

The theory of groups, an introduction, 2nd edition,

Allyn and Bacon, Boston, 1973. 21.

J. Stallings.

Group theory and three-dimensional manifolds,

Yale University Press, New Haven, 1971. 22.

J. W. Wamsley.

Minimal presentations for finite groups, Bull.

London Math. Soc. , 5 (1973), 129-44.

197

Index of notation

N

natural numbers

z

integers, or cyclic group of infinite order

Q

rationals

R

reals

c

complex numbers

n

intersection

u

union

u sn zn

disjoint union symmetric group of degree n e: N u { oo) cyclic group of order n

(X/R)

group with generators X and relations (relators) R

subgroup generated by X

c;;;

set- theoretic containment

c

proper set-theoretic containment

:$

containment of subgroup, or 'less than or equal'