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Lecture Notes inMathematicsA collection of informal reports and seminars

Edited by A. Dold, Heidelberg and B. Eckmann, ZOrich

143

Karl W. GruenbergQueen Mary College, London

Cohomological Topicsin Group Theory

Springer-VerlagBerlin' Heidelberg ·NewYork 1970

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@ by Springer-Yerlag Berlin' Heidelberg 1970. Library of Congress Catalog Card Number 70-12700 Printed in Germany.TItle No. 3299

PREFACE

These notes are based on lectures that I have given at

various times during the last four years and at various places,

but mainly at Queen Mary College, London. Chapters 1 to 7 have

been in ciroulation as a volume in the Queen Mary College

Mathematios Notes since the autumn of 1967. They are reproduoed

here unchanged exoept for the addition of some bibliographioal

material and the oorreotion of some minor errors.

Chapter 8 is an attempt at a reasonably oomplete survey

of the subjeot of finite oohomologioal dimension. I have

included proofs of everything that is not readily acoessible

in the literature.

Chapters 9 and 11 oontain an aooount of a kind of

globalised extension theory whioh I believe to be new. A survey

of some of the results has appeared in volume 2 of "Category

theory, homology theory and their applioations", Springer

Leoture Notes, no.92 (1969). The basic maohinery of extension

oategories for arbitrary groups is given in ohapter 9. Then

in chapter 11 we focus attention exolusively on finite groups

and primarily on the structure of minimal projective extensions.

Chapter 10 is purely auxiliary and merely sets out some

cohomological facts needed in chapter 11.

My aim in these lectures was to present cohomology as a

tool for the study of groups. In this respect they differ

basioally from other available acoounts of group cohoaology in

iv

all of which the theory is developed with an eye on arithmetical

applications. Our subject here is group theory with a

cohomo1ogical flavour.

It should be stressed that there is no pretence whatsoever

at completeness. In fact, the general homological machinery is

kept to the bare minimum needed for the topics' at hand. It

follows - inevitably - that many important features are barely

mentioned; and Some not at all.

The audiences were not assumed to know anything about

homological algebra except the most rUdimentary facts. A little

more knowledge of group theory was presupposed, but nothing at

all sophisticated. Full references to all non-trivial or

non-standard results are always given.

There is a list of the most frequently quoted books

immediately following this preface. Each chapter ends with a

list of all articles and books mentioned in that chapter and

reference numbers refer to that list at the end of the chapter

where they occur.

I was fortunate to have perceptive audiences who frequently

saved me from errore and obscurities. My thanks go to all who

participated and in particular to D. Cohen, I. Kap1ansky,

D. Knudson, A. Learner, H. Mochizuki, G. Rinehart, W. Vasconcelos

and B. Wehrfritz. lowe a speoia1 debt of gratitude to

Urs Stammbach for his oareful and critioa1 reading of large

sections of these notes.

v

I am also grateful to Cornell University, the University

of Oregon, the University of British Columbia and the Eidgen.

Tech. Hochschule, ZUrich, for financial assistance at various

stages of this work.

The notes were typed by Mrs. Esther Monroe and

Miss Valerie Kinsella and I thank them both for their enormous

patience with me and their excellent work.

Queen Mary College,

London,

February 1970.

CONTENTS

Preface

Book list

Leitfaden

Some notation and terminology

iii

xi

xii

xiii

CHAPTER 1: Fixed point free action 1

1.1 The fixed point functor and its dual 1

1.2 Elementary consequences of fixed point free action :5

1.; Finite groups 5

Sources and references 12

CHAPTER 2: The cohomology and homology groups 15

The cohomology functor

The homology functor

Change of coefficient ring

Isomorphism of group rings

Sources and references

CHAPTER;: Presentations and resolutions

A functor from presentations to resolutions

Remarks on the construction of

Cyclic groups

The standard resolution

15

21

25

26

29

6.1

6.2

B1(G, ) and Bl(G, )

B2(G, ) and H2(G, )

The universal coefficient theorem for cohomology

Referenoes

CHAPTER 4: Free groups

Dimension subgroups

Residual nilpotence of the augmentation ideal

Residual properties of free groups

Power series

Units and zero divisors

Souroes and references

CHAPTER 5: Classical extension theory

The problem

Covering groups

Extensions with abelian kernel

General extensions

Obstruotions

Sources and references

CHAPTER 6: More oohomological machinery

Batural homomorphisms of cohomological functors

Restriction, inflation, oorestriction

vii

44

46

48

50

51

51

54

57

59

61

63

65

65

67

70

73

76

84

85

85

88

viii

8.1

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8.9

8.10

8.11

8.12

The Shapiro lemma

The inflation-restriction sequence

The trace map for finite groups

CHAPTER 7: Finite p-groups

Frattini groups

Generators and relations for p-groups..The Golod-Safarevic inequality

Hilbert class fields

Outer automorphisms of order p

Sources and references

CHAPTER 8: Cohomological dimension

Definition and elementary facts

Test elements

Some groups of cohomo1ogica1 dimension 2

One relator groups

Direct limits

Free products

Extensions

Nilpotent groups

Centres

Euler characteristics

Trivial cohomo1ogica1 dimension

Finite groups

Sources and references

91

93

94

97

97

99

104

107

110

116

119

119

122

125

129

132

138

145

148

155

159

168

175

179

ix

CHAPTER 9: Extension categories: general theory 185

9.1 The categories (Q:.) and9.2 Two theorems of Schur

9.3 Monomorphisms and epimorphisms

9.4 Injective objects

9.5 Projective objects

9.6 Minimal projectives

9.7 Change of coefficient ring

9.8 Projective covers

9.9 Central extensions

Sources and references

CHAPTER 10: More module theory

10.1 Module extensions

10.2 Heller modules

10.3 Ext under flat coefficient extensions

10.4 Localisation

10.5 Local rings

10.6 Semi-local rings

10.7 Cohomo1ogical criteria for projectivity

Sources and references

185

189

191

194

196

201

204

206

210

218

221

221

227

230

234

238

240

242

247

CHAPTER 11: Extension categories: finite groups 249

11.1 Minimal projectives when IGI is invertible in K 250

x

11.2

11.3

11.4

11.5

11.6

11.7

11.8

Existence of projeotive covers

Oohamologioal properties of projectives

OohOlllological oharaoter18atlon of projeotives

Uniqueaess of minimal projeotives

Minimal free extensiollS

The module struoture of minimal projeotlves

Oonolusion

Souroes and references

251

255

258

262

267

270

273

274

BOOK LIST

The following books are usually referred to by their

author's name only.

Burnside, W.: The theory of groupe of finite order, Cambridge,

2nd edition, 1911 (Chelsea 1958).

Cartan, H. and Eilenberg, S.: Homological algebra, Princeton

1956.

Curtis, C.W. and Reiner, I.: Representation theory of finite

groups and associative algebras, Interscience,

1962.

Hall, P.: Nilpotent groups, Notes of lectures at the Canadian

Mathematical Congress, Univ. of Alberta, 1957.

(Reprinted: Queen Mary College Mathematics Notes,

1969).

Huppert, B.: Endliche Gruppen I, Springer, 1967.

Lang, S.: Rapport sur la cohomologie des groupes, Benjamin, 1966.

Rotman, J.: The theory of groups: an introduction, Allyn and

Beacon, 1965.

Schenkmah, E.: Group theory, van Nostrand, 1965.

Scott, W.R.: Group theory, Prentice-Hall, 1964.

Serre, J.-P.: Corps Locaux, Hermann, 1962.

"UNGEFAHRER LEITFADEN.

2

11

SOME NOTATION AND TERMINOLOGY

Let G be a group.

If S is a subset of a G-group M (p.l),

xiv

If H, K are subgroups of G and K H (normal), H/K is a faotor of G.

If (H,G] K, the factor is called oentral.

A finite series is a family of subgroups (3 i ; 0 i m), where

3 i 3 i +l•If all factors are oentral, the series is oalled a oentral series.

If G has a finite central series from 1 to G (i.e., So = 1 and3m = G), then G is called nilpotent.If G has a finite series from 1 to G with all factors abelian

(oyclic), then G is called soluble (polycyclic).

h(G) = Hirsch number of the locally polycyclic G (§8.8).

If 'o(G) =1, 'l(G) = centre of G, and 'k+l(G) is the uniquesubgroup so that 'k+l(G)/(k(G) = 'l(G/'k(G», then ('i(G);i 0)is called the upper central series of G.

If G is nilpotent and 'C_l(G) < 'c(G) = G, then c is theof G.

If Gl = G, Gk+l = (Gk,G], then (Gt ; i 1) is called thecentral series of G.

If G(O) = G, G' = (G,G] (= G2)

and G(m+l) = (G(m»" then(G(i); i 0) is called the derived series of G.