Cohomology of Finite Groups

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 This work is subject to copyright. All rights are reserved, whether the whole or part ofthe material is concemed, specifically the rights oftranslation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from SpringerVerlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Originally published by Springer-Verlag Berlin Heidelberg New York in 1994. Softcover reprint of the hardcover 1st edition 1994 Typesetting: Camera-ready copy produced by the authors' output file using aSpringer TEX macro package 41/3140-54321 0 Printed on acid-free paper SPIN 10078665 Table of Contents Introduction ................................................ 1 Chapter I. Group Extensions, Simple Aigebras and Cohomology o. 1. 2. 3. 4. 5. 6. 7. 8. Introduction .............................................. Group Extensions ......................................... Extensions Associated to the Quaternions .................... The Group of Unit Quaternions and SO(3) ................... The Generalized Quaternion Groups and Binary Tetrahedral Group ................................................... Central Extensions and SI Bundles on the Torus T 2 ••..••.•.•• The Pull-back Construction and Extensions .................. The Obstruction to Extension When the Center Is Non-Trivial .. Counting the Number of Extensions ......................... The Relation Satisfied by JL(gI, g2, g3) ....................... A Certain Universal Extension .............................. Each Element in H~(G; C) Represents an Obstruction ......... Associative Aigebras and H~(G; C) .......................... Basic Structure Theorems for Central Simple lF-Algebras ....... Tensor Products of Central Simple lF-Algebras ................ The Cohomological Interpretation of Central Simple Division Aigebras ................................................. Comparing Different Maximal Subfields, the Brauer Group ..... 7 8 12 14 16 18 20 23 27 32 34 35 36 36 38 40 43 Chapter 11. Classifying Spaces and Group Cohomology O. 1. 2. 3. 4. Introduction .............................................. Preliminaries on Classifying Spaces .......................... Eilenberg-MacLane Spaces and the Steenrod Algebra A(p) ..... Axioms for the Steenrod Algebra A(2) ....................... Axioms for the Steenrod Algebra A(p) ....................... The Cohomology of Eilenberg-MacLane Spaces ................ The Hopf Algebra Structure on A(p) ........................ Group Cohomology ........................................ Cup Products ............................................. 45 45 53 55 55 56 57 57 66 VI 5. 6. 7. 8. Table of Contents Restrietion and Thansfer Thansfer and Restrietion for Abelian Groups .................. An Alternate Construction of the Thansfer .................... The Cartan-Eilenberg Double Coset Formula ................. Tate Cohomology and Applications .......................... The First Cohomology Group and Out(G) .................... 69 71 73 76 81 87 Chapter 111. Modular Invariant Theory O. 1. 2. 3. 4. 5. 6. Introduction .............................................. Generallnvariants............................ . . . . . . . . . . . . . The Dickson Algebra ...................................... A Theorem of Serre ....................................... The Invariants in H*((Zjp)nj'Fp ) Under the Action of Sn The Cardenas-Kuhn Theorem... ........ .............. .... .. Discussion of Related Topics and Further Results .............. The Diekson Aigebras and Topology ......................... The Ring of Invariants for SP2n('F 2) ......................... The Invariants of Subgroups of GL4('F2) ...................... 93 93 100 105 108 112 115 115 115 116 Chapter IV. Spectral Sequences and Detection Theorems O. 1. 2. 3. 4. 5. 6. 7. Introduction .............................................. The Lyndon-Hochschild-Serre Spectral Sequence: Geometrie Approach ................................................ Wreath Products .......................................... Central Extensions ........................................ A Lemma of Quillen-Venkov ................................ Change of Rings and the Lyndon-Hochschild-Serre Spectral Sequence ................................................. The Dihedral Group D 2n ................................... The Quaternion Group Qs ................................. Chain Approximations in Acyclie Complexes .................. Groups With Cohomology Detected by Abelian Subgroups ..... Structure Theorems for the Ring H* (Gj 'Fp) .................. Evens-Venkov Finite Generation Theorem .................... The Quillen-Venkov Theorem ............................... The Krull Dimension of H* (Gj 'F p) ......... . . . . . . . . . . . . . . . .. The Classification and Cohomology Rings of Periodie Groups ... The Classification of Periodie Groups ........................ The Mod(2) Cohomology of the Periodic Groups .............. The Definition and Properties of Steenrod Squares ......... . .. The Squaring Operations ................................... The P-Power Operations for p Odd .......................... 117 118 119 122 124 125 128 131 134 140 143 143 144 144 146 149 154 156 157 159 Table of Contents VII Chapter V. G-Complexes and Equivariant Cohomology O. 1. 2. 3. Introduction to Cohomological Methods ...................... Restrietions on Group Actions .............................. General Properties of Posets Associated to Finite Groups ...... Applications to Cohomology ................................ S4 ....................................................... SL3 (lF 2 ) ••••••••••••••••••••••••••••••••.•••••••••••••••.• The Sporadic Group Mn ................................... The Sporadic Group J 1 .•.•.•.•....•..•••.••.••••••••.••..• 161 165 170 176 178 178 179 179 Chapter VI. The Cohomology of Symmetrie Groups O. 1. 2. 3. 4. 5. 6. Introduction .............................................. Detection Theorems for H*(Snj lFp ) and Construction of Generators ............................................... Hopf Algebras ............................................ The Theorems of Borel and Hopf ............................ The Structure of H*(SnjlF p ) •••••••.•••••••••••••••••••••••• More Invariant Theory ..................................... H*(Sn), n = 6,8,10,12 ................................... The Cohomology of the Alternating Groups .................. 181 184 197 201 203 206 211 214 Chapter VII. Finite Groups of Lie Type 1. 2. 3. 4. 5. 6. 7. Preliminary Remarks ...................................... The Classical Groups of Lie Type ........................... The Orders of the Finite Orthogonal and Symplectic Groups .... The Cohomology of the Groups GLn(q) ...................... The Cohomology of the Groups O;'(q) for q Odd .............. The Cohomology Groups H*(Om(q)jlF 2) ..................... The Groups H*(SP2n(q)jlF 2) ................................ The Exceptional Chevalley Groups .......................... 219 220 227 231 235 240 241 246 Chapter VIII. Cohomology of Sporadie Simple Groups O. 1. 2. 3. 4. 5. Introduction .............................................. The Cohomology of Mn .................................. , The Cohomology of J 1 ••••••••••••.•...••..••••••.••..•.••. The Cohomology of M 12 ................................... The Structure of Mathieu Group M 12 •..•••••••.•.••••..••••• The Cohomology of M 12 ................................... Discussion of H*(M12jlF2) .................................. The Cohomology of Other Sporadic Simple Groups ............ The O'Nan Group 0' N .................................... 251 252 253 254 254 258 263 267 267 VIII Table of Contents The Mathieu Group M 22 The Mathieu Group M 23 268 271 Chapter IX. The Plus Construction and Applications O. 1. 2. 3. 4. Preliminaries ............................................. Definitions ............................................... Classification and Construction of Acyclic Maps ............... Examples and Applications ................................. The Infinite Symmetrie Group .............................. The General Linear Group Over a Finite Field ................ The Binary Icosahedral Group .............................. The Mathieu Group M 12 •••••••••••••••••••••••••••••••.••• The Group J 1 ••.••..••••.• • . • . • • . . • • . . • • . • • • . • • • • . • • • • • .. The Mathieu Group M 23 ••••..•••••••••.••••••••••.•••..••• The Kan-Thurston Theorem ...... . . . . . . . . . . . . . . . . . . . . . . . .. 273 273 275 277 277 278 279 281 281 282 283 Chapter X. The Schur Subgroup of the Brauer Group O. 1. 2. 3. 4. 5. Introduction .............................................. The Brauer Groups of Complete Local Fields ................. Valuations and Completions ................................ The Brauer Groups of Complete Fields with Finite Valuations " The Brauer Group and the Schur Subgroup for Finite Extensions of Q ........................................... The Brauer Group of a Finite Extension of Q ................. The Schur Subgroup of the Brauer Group .................... The Group (QjZ) and Its Aut Group ........................ The Explicit Generators of the Schur Subgroup ............... Cyclotomic Algebras and the Brauer-Witt Theorem ........... The Galois Group of the Maximal Cyclotomie Extension of lF ... The Cohomologieal Reformulation of the Schur Subgroup ... . .. The Groups H;ont(GFiQjZ) and H;ont(GviQjZ) .............. The Cohomology Groups H;ont(GFi QjZ) .................... The Local Cohomology with QjZ Coefficients ................. The Explicit Form of the Evaluation Maps at the Finite Valuations ................................................ The Explicit Structure of the Schur Subgroup, S(lF) ........... The Map H;ont(Gvi QjZ)---tH;ont(Gvi Q;,cycl)' ................ The Invariants at the Infinite Real Primes .................... The Remaining Local Maps ................................. 289 290 290 293 295 295 297 298 299 299 300 301 304 304 307 309 310 311 314 316 References .................................................. 319 Index ....................................................... 325