Cohomology of finite groups

Table of Contents......Page 3
Introduction......Page 7
0. Introduction......Page 12
1. Group Extensions......Page 13
2. Extensions Associated to the Quaternions......Page 17
The Group of Unit Quaternions and SO(3)......Page 19
The Generalized Quaternion Groups and Binary Tetrahedral Group......Page 21
3. Central Extensions and S^1 Bundles on the Torus T^2......Page 23
4. The Pull-back Construction and Extensions......Page 25
5. The Obstruction to Extension When the Center Is Non-Trivial......Page 28
6. Counting the Number of Extensions......Page 32
7. The Relation Satisfied by \\mu(g_1, g_2, g_3)......Page 37
A Certain Universal Extension......Page 39
Each Element in H^3_\\Phi(G; C) Represents an Obstruction......Page 40
Basic Structure Theorems for Central Simple F-Algebras......Page 41
Tensor Products of Central Simple F-Algebras......Page 43
The Cohomological Interpretation of Central Simple Division Algebras......Page 45
Comparing Different Maximal Subfields, the Brauer Group......Page 48
1. Preliminaries on Classifying Spaces......Page 50
2. Eilenberg-MacLane Spaces and the Steenrod Algebra A(p)......Page 58
Axioms for the Steenrod Algebra A(p)......Page 60
The Cohomology of Eilenberg-MacLane Spaces......Page 61
3. Group Cohomology......Page 62
4. Cup Products......Page 71
5. Restriction and Transfer......Page 74
Transfer and Restriction for Abelian Groups......Page 76
An Alternate Construction of the Transfer......Page 78
6. The Cartan-Eilenberg Double Coset Formula......Page 81
7. Tate Cohomology and Applications......Page 86
8. The First Cohomology Group and Out(G)......Page 92
1. General Invariants......Page 98
2. The Dickson Algebra......Page 105
3. A Theorem of Serre......Page 110
4. The Invariants in H^*((Z/p)^n; F_p) Under the Action of S_n......Page 113
5. The Cardenas-Kuhn Theorem......Page 117
The Ring of Invariants for Sp_{2n}(F_2)......Page 120
The Invariants of Subgroups of GL_4(F_2)......Page 121
0. Introduction......Page 122
1. The Lyndon-Hochschild-Serre Spectral Sequence: Geometric Approach......Page 123
Wreath Products......Page 124
Central Extensions......Page 127
A Lemma of Quillen-Venkov......Page 129
2. Change of Rings and the Lyndon-Hochschild-Serre Spectral Sequence......Page 130
The Dihedral Group D_{2n}......Page 133
The Quaternion Group Q_8......Page 136
3. Chain Approximations in Acyclic Complexes......Page 139
4. Groups With Cohomology Detected by Abelian Subgroups......Page 145
Evens-Venkov Finite Generation Theorem......Page 148
The Krull Dimension of H^*(G; F_p)......Page 149
6. The Classification and Cohomology Rings of Periodic Groups......Page 151
The Classification of Periodic Groups......Page 154
The Mod(2) Cohomology of the Periodic Groups......Page 159
7. The Definition and Properties of Steenrod Squares......Page 161
The Squaring Operations......Page 162
The P-Power Operations for p Odd......Page 164
0. Introduction to Cohomological Methods......Page 165
1. Restrictions on Group Actions......Page 169
2. General Properties of Posets Associated to Finite Groups......Page 174
3. Applications to Cohomology......Page 180
SL_3(F_2)......Page 182
The Sporadic Group J_1......Page 183
0. Introduction......Page 184
1. Detection Theorems for H^*(S_n; F_p) and Construction of Generators......Page 187
2. Hopf Algebras......Page 200
The Theorems of Borel and Hopf......Page 204
3. The Structure of H_*(S_n; F_p)......Page 206
4. More Invariant Theory......Page 209
5. H^*(S_n), n = 6, 8, 10, 12......Page 214
6. The Cohomology of the Alternating Groups......Page 217
1. Preliminary Remarks......Page 222
2. The Classical Groups of Lie Type......Page 223
3. The Orders of the Finite Orthogonal and Symplectic Groups......Page 230
4. The Cohomology of the Groups GL_n(q)......Page 234
5. The Cohomology of the Groups O^*_n(q) for q Odd......Page 238
The Cohomology Groups H^*(O_m(q); F_2)......Page 243
6. The Groups H^*(Sp_{2n}(q); F_2)......Page 244
7. The Exceptional Chevalley Groups......Page 249
0. Introduction......Page 254
1. The Cohomology of M_{11}......Page 255
2. The Cohomology of J_1......Page 256
The Structure of Mathieu Group M_{12}......Page 257
The Cohomology of M_{12}......Page 261
4. Discussion of H^*(M_{12}; F_2)......Page 266
The O\'Nan Group O\'N......Page 270
The Mathieu Group M_{22}......Page 271
The Mathieu Group M_{23}......Page 274
1. Definitions......Page 276
2. Classification and Construction of Acyclic Maps......Page 278
The Infinite Symmetric Group......Page 280
The General Linear Group Over a Finite Field......Page 281
The Binary Icosahedral Group......Page 282
The Group J_1......Page 284
The Mathieu Group M_{23}......Page 285
4. The Kan-Thurston Theorem......Page 286
0. Introduction......Page 291
Valuations and Completions......Page 292
The Brauer Groups of Complete Fields with Finite Valuations......Page 295
The Brauer Group of a Finite Extension of Q......Page 297
The Schur Subgroup of the Brauer Group......Page 299
The Group (Q/Z) and Its Aut Group......Page 300
Cyclotomic Algebras and the Brauer-Witt Theorem......Page 301
The Galois Group of the Maximal Cyclotomic Extension of F......Page 302
The Cohomological Reformulation of the Schur Subgroup......Page 303
The Cohomology Groups H^*_{cont}(G_F; Q/Z)......Page 306
The Local Cohomology with Q/Z Coefficients......Page 309
The Explicit Form of the Evaluation Maps at the Finite Valuations......Page 311
5. The Explicit Structure of the Schur Subgroup, S(F)......Page 312
The Map H^*_{cont}(G_v; Q/Z) -> H^2_{cont}(G_v; \\hat{Q}^*_{p,cycl})......Page 313
The Invariants at the Infinite Real Primes......Page 316
The Remaining Local Maps......Page 318
References......Page 320
Index......Page 326