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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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Under § 54 of the German Copyright Lawwhere copies are made for other than private use, a fee is payable to the publisher,the amount of the fee to be determined by agreement with the publisher.
@ by Springer-Yerlag Berlin' Heidelberg 1970. Library of Congress Catalog Card Number 70-12700 Printed in Germany.TItle No. 3299
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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
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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.
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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.
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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
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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
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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
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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
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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
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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.
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"UNGEFAHRER LEITFADEN.
2
11
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SOME NOTATION AND TERMINOLOGY
Let G be a group.
If S is a subset of a G-group M (p.l),
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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.