Cohomology of finite groups

333 Pages·1994·25.381 MB·English

by  Alejandro Adem, R James Milgram

Grundlehren der
mathematischen Wissenschaften 309
ASeries 0/ Comprehensive Studies in Mathematics
Editors
M. Artin S. S. Chern 1. Coates 1. M. Fröhlich
H. Hironaka F. Hirzebruch L. Hörmander
C. C. Moore 1. K. Moser M. Nagata W. Schmidt
D. S. Scott Ya. G. Sinai 1. Tits M. Waldschmidt
S.Watanabe
Managing Editors
M. Berger B. Eckmann S. R. S. Varadhan
Alejandro Adern R. James Milgram
Cohomology
of Finite Groups
Springer-Verlag Berlin Heidelberg GmbH
Alejandro Adern
Department of Mathernatics
University of Wisconsin
Madison, Wl 53706, USA
R. James Milgram
Departrnent of Applied Hornotopy
Stanford University
Stanford, CA 94305-9701, USA
Mathernatics Subject Classification (1991): 20J05, 20J06, 20110,
55R35, 55R40, 57S17, 18GlO, 18G15, 18G20, 18G40
ISBN 978-3-662-06284-5 ISBN 978-3-662-06282-1 (eBook)
DOI 10.1007/978-3-662-06282-1
Library of Congress Cataloging-in-Publication Data
Adern, Alejandro. Cohomology of finite groupsl
Alejandro Adern, Richard James Milgram.
p. cm. - (Grundlehren der mathematischen Wissenschaften; 309)
Inciudes bibliographical references and index.
1. Finite groups. 2. Homology theory. I. Milgram, R. James. 11. Title. 111. Series.
QA177.A34 1995 512'.55-dc20 94-13318 CIP
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Originally published by Springer-Verlag Berlin Heidelberg New York in 1994.
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Table of Contents
Introduction ................................................ 1
Chapter I. Group Extensions, Simple Aigebras
and Cohomology
o.
Introduction .............................................. 7
1. Group Extensions ......................................... 8
2. Extensions Associated to the Quaternions .................... 12
The Group of Unit Quaternions and SO(3) ................... 14
The Generalized Quaternion Groups and Binary Tetrahedral
Group ................................................... 16
3. Central Extensions and SI Bundles on the Torus T2 ••..••.•.•• 18
4. The Pull-back Construction and Extensions .................. 20
5. The Obstruction to Extension When the Center Is Non-Trivial .. 23
6. Counting the Number of Extensions ......................... 27
7. The Relation Satisfied by JL(gI, g2, g3) ....................... 32
A Certain Universal Extension .............................. 34
Each Element in H~(G; C) Represents an Obstruction ......... 35
8. Associative Aigebras and H~(G; C) .......................... 36
Basic Structure Theorems for Central Simple lF-Algebras ....... 36
Tensor Products of Central Simple lF-Algebras ................ 38
The Cohomological Interpretation of Central Simple Division
Aigebras ................................................. 40
Comparing Different Maximal Subfields, the Brauer Group ..... 43
Chapter 11. Classifying Spaces and Group Cohomology
O. Introduction .............................................. 45
1. Preliminaries on Classifying Spaces .......................... 45
2. Eilenberg-MacLane Spaces and the Steenrod Algebra A(p) ..... 53
Axioms for the Steenrod Algebra A(2) ....................... 55
Axioms for the Steenrod Algebra A(p) ....................... 55
The Cohomology of Eilenberg-MacLane Spaces ................ 56
The Hopf Algebra Structure on A(p) ........................ 57
3. Group Cohomology ........................................ 57
4. Cup Products ............................................. 66
VI Table of Contents
5. Restrietion and Thansfer 69
Thansfer and Restrietion for Abelian Groups .................. 71
An Alternate Construction of the Thansfer .................... 73
6. The Cartan-Eilenberg Double Coset Formula ................. 76
7. Tate Cohomology and Applications .......................... 81
8. The First Cohomology Group and Out(G ) .................... 87
Chapter 111. Modular Invariant Theory
O. Introduction .............................................. 93
1. Generallnvariants............................ . . . . . . . . . . . . . 93
2. The Dickson Algebra ...................................... 100
3. A Theorem of Serre ....................................... 105
4. The Invariants in H*((Zjp)nj'Fp) Under the Action of Sn 108
5. The Cardenas-Kuhn Theorem... ........ .............. .... .. 112
6. Discussion of Related Topics and Further Results .............. 115
The Diekson Aigebras and Topology ......................... 115
The Ring of Invariants for SP2n('F2) ......................... 115
The Invariants of Subgroups of GL4('F2) ...................... 116
Chapter IV. Spectral Sequences and Detection Theorems
O. Introduction .............................................. 117
1. The Lyndon-Hochschild-Serre Spectral Sequence: Geometrie
Approach ................................................ 118
Wreath Products .......................................... 119
Central Extensions ........................................ 122
A Lemma of Quillen-Venkov ................................ 124
2. Change of Rings and the Lyndon-Hochschild-Serre Spectral
Sequence ................................................. 125
The Dihedral Group D2n ................................... 128
The Quaternion Group Qs ................................. 131
3. Chain Approximations in Acyclie Complexes .................. 134
4. Groups With Cohomology Detected by Abelian Subgroups ..... 140
5. Structure Theorems for the Ring H* (Gj 'F p) .................. 143
Evens-Venkov Finite Generation Theorem .................... 143
The Quillen-Venkov Theorem ............................... 144
The Krull Dimension of H* (Gj 'F p) ......... . . . . . . . . . . . . . . . .. 144
6. The Classification and Cohomology Rings of Periodie Groups ... 146
The Classification of Periodie Groups ........................ 149
The Mod(2) Cohomology of the Periodic Groups .............. 154
7. The Definition and Properties of Steenrod Squares ......... . .. 156
The Squaring Operations ................................... 157
The P-Power Operations for p Odd .......................... 159
Table of Contents VII
Chapter V. G-Complexes and Equivariant Cohomology
O. Introduction to Cohomological Methods ...................... 161
1. Restrietions on Group Actions .............................. 165
2. General Properties of Posets Associated to Finite Groups ...... 170
3. Applications to Cohomology ................................ 176
S4 ....................................................... 178
SL3(lF2) ••••••••••••••••••••••••••••••••.•••••••••••••••.• 178
The Sporadic Group Mn ................................... 179
The Sporadic Group J1 .•.•.•.•....•..•••.••.••••••••.••..• 179
Chapter VI. The Cohomology of Symmetrie Groups
O. Introduction .............................................. 181
1. Detection Theorems for H*(Snj lFp) and Construction of
Generators ............................................... 184
2. Hopf Algebras ............................................ 197
The Theorems of Borel and Hopf ............................ 201
3. The Structure of H*(SnjlFp) •••••••.•••••••••••••••••••••••• 203
4. More Invariant Theory ..................................... 206
5. H*(Sn), n = 6,8,10,12 ................................... 211
6. The Cohomology of the Alternating Groups .................. 214
Chapter VII. Finite Groups of Lie Type
1. Preliminary Remarks ...................................... 219
2. The Classical Groups of Lie Type ........................... 220
3. The Orders of the Finite Orthogonal and Symplectic Groups .... 227
4. The Cohomology of the Groups GLn(q) ...................... 231
5. The Cohomology of the Groups O;'(q) for q Odd .............. 235
The Cohomology Groups H*(Om(q)jlF2) ..................... 240
6. The Groups H*(SP2n(q)jlF2) ................................ 241
7. The Exceptional Chevalley Groups .......................... 246
Chapter VIII. Cohomology of Sporadie Simple Groups
O. Introduction .............................................. 251
1. The Cohomology of Mn .................................. , 252
2. The Cohomology of J1 ••••••••••••.•...••..••••••.••..•.••. 253
3. The Cohomology of M12 ................................... 254
The Structure of Mathieu Group M12 •..•••••••.•.••••..••••• 254
The Cohomology of M12 ................................... 258
4. Discussion of H*(M12jlF2) .................................. 263
5. The Cohomology of Other Sporadic Simple Groups ............ 267
The O'Nan Group 0' N .................................... 267
VIII Table of Contents
The Mathieu Group M22 268
The Mathieu Group M23 271
Chapter IX. The Plus Construction and Applications
O. Preliminaries ............................................. 273
1. Definitions ............................................... 273
2. Classification and Construction of Acyclic Maps ............... 275
3. Examples and Applications ................................. 277
The Infinite Symmetrie Group .............................. 277
The General Linear Group Over a Finite Field ................ 278
The Binary Icosahedral Group .............................. 279
The Mathieu Group M12 •••••••••••••••••••••••••••••••.••• 281
The Group J1 ••.••..••••.• • . • . • • . . • • . . • • . • • • . • • • • . • • • • • .. 281
The Mathieu Group M23 ••••..•••••••••.••••••••••.•••..••• 282
4. The Kan-Thurston Theorem ...... . . . . . . . . . . . . . . . . . . . . . . . .. 283
Chapter X. The Schur Subgroup of the Brauer Group
O. Introduction .............................................. 289
1. The Brauer Groups of Complete Local Fields ................. 290
Valuations and Completions ................................ 290
The Brauer Groups of Complete Fields with Finite Valuations " 293
2. The Brauer Group and the Schur Subgroup for Finite
Extensions of Q ........................................... 295
The Brauer Group of a Finite Extension of Q ................. 295
The Schur Subgroup of the Brauer Group .................... 297
The Group (QjZ) and Its Aut Group ........................ 298
3. The Explicit Generators of the Schur Subgroup ............... 299
Cyclotomic Algebras and the Brauer-Witt Theorem ........... 299
The Galois Group of the Maximal Cyclotomie Extension of lF ... 300
The Cohomologieal Reformulation of the Schur Subgroup ... . .. 301
4. The Groups H;ont(GFiQjZ) and H;ont(GviQjZ) .............. 304
The Cohomology Groups H;ont(GFi QjZ) .................... 304
The Local Cohomology with QjZ Coefficients ................. 307
The Explicit Form of the Evaluation Maps at the Finite
Valuations ................................................ 309
5. The Explicit Structure of the Schur Subgroup, S(lF) ........... 310
The Map H;ont(Gvi QjZ)---tH;ont(Gvi Q;,cycl)' ................ 311
The Invariants at the Infinite Real Primes .................... 314
The Remaining Local Maps ................................. 316
References .................................................. 319
Index ....................................................... 325
Introduction
Some Historical Background
This book deals with the cohomology of groups, particularly finite ones. His
torically, the subject has been one of significant interaction between algebra
and topology and has directly led to the creation of such important areas of
mathematics as homological algebra and algebraic K-theory. It arose primar
ily in the 1920's and 1930's independently in number theory and topology.
In topology the main focus was on the work of H. Hopf, but B. Eckmann, S.
Eilenberg, and S. MacLane (among others) made significant contributions.
The main thrust of the early work here was to try to understand the mean
ings of the low dimensional homology groups of aspace X. For example, if
the universal cover of X was three connected, it was known that H2(X;A)
depends only on the fundamental group of X. Group cohomology initially
appeared to explain this dependence.
In number theory, group cohomology arose as a natural device for describ
ing the main theorems of class field theory and, in particular, for describing
and analyzing the Brauer group of a field. It also arose naturally in the study
of group extensions, N<l E-+G, where H3(G; C) carries the obstruction to
the existence of any extension at all, H2(G; C) counts the number of distinct
extensions. Here C is the center of N.
In these original algebraic applications the emphasis was on cohomology
with twisted coefficients, Le., where the group acts non-trivially on the co
efficient group. For example, the action in H* (G; C) is that induced from
the action of E on N by conjugation. On the other hand, in topology the
emphasis was almost exclusively on the situation where the action is triv
ial. These basic applications are discussed in Chapter I which is, to a large
degree, a historical introduction to the subject, concentrating on the exten
sion problem for groups and the description of the Brauer group in terms of
group cohomology. In particular it develops the extension theory for groups
and central simple algebras to motivate the definition of group cohomology.
Finite groups and their cohomology also arise in myriad other contexts in
topology and algebra. One of the classic successes in the area was the proof
(due to P. Smith) that any finite group which acts freely on a sphere must
be periodic (equivalently, have Krull dimension one, Le., have all its abelian
subgroups cyclic). Such groups have been classified. They are discussed in
2 Introduction
(IV.6). The simplest of them is (Zj2) and its classifying space, the infinite
real projective space JRlP>00, was the main tool used by J.F. Adams in solving
the problem of vector fields on spheres. Likewise, Quillen used the structure
of the cohomology of the finite groups of Lie type (Chapter VII) to prove
the Adams conjecture identifying the im( J) groups as direct summands of
the stable homotopy groups of spheres. More recently, with the proof by G.
Carlsson of the Segal conjecture, it is evident that the dominant influence on
the structure of stable homotopy theory is contained in the structure of the
finite symmetrie groups.
From a more algebraic point of view the cohomology ring H* (G; OC) of
a finite group with coefficients in a finite field is connected via work of D.
Quillen, J. Alperin, L. Evens, J. Carlson, and D. Benson to the structure of
the modular representations of G. This connection and its ramifications have
provided a vast increase in interest in the subject among algebraists. The
theoretieal underpinnings here have been discussed at length in the excellent
books ofD. Benson, [Be), and L. Evens, [Ev2) , so we do not repeat them here.
What we do is to supply the techniques and examples needed to flesh out
the theory. For example see Chapter VIII where we discuss the cohomology
of some of the sporadie groups, notably the Mathieu groups Mn, M 12, M22
and the group 0'N , and Chapter VII where we discuss the finite Chevalley
groups.
In topology the main source of examples and test spaces are classifying
spaces of groups, various natural subspaces, and maps induced by homomor
phisms of the groups. This is hardly suprising since the Kan-Thurston theo
rem and Quillen's plus construction (both discussed in Chapter IX) show that
any simply connected space can be constructed from the classifying spaces of
groups in very simple ways. For example the plus construction on the classi
fying space of the infinite symmetrie group, 500, is identified with the space
limn [ln sn, the infinite loop space of the infinite sphere. Also, the groups of
Lie type over finite fields lead to models for Bo, Bu, Bs etc.
p,
Related to this, and a major motivation for the study of group cohomology
in algebraic topology was Steenrod's construction of cohomology operations
in arbitrary topologieal spaces using the eohomology of the symmetrie groups.
(This construction is reviewed in (IV. 7) and the eohomology of the symmetrie
groups is discussed in Chapter VI.)
Another reason for the significance of group cohomology sterns from its
direet relationship with both group actions and homotopy theory. Methods
developed by P. Smith to study finite group actions uncovered substantial
cohomologieal restrietions in transformation groups (Chapter V) whieh led
A. Borel to develop a systematie method for analyzing group actions.