Cohomological Methods in Group Theory

Cohomological Methods in Group Theory......Page 1
ISBN 0824710312......Page 4
Preface......Page 5
Contents......Page 7
1. Modules......Page 11
2. Tensor Products......Page 13
3. Existence of Tensor Products......Page 14
4. The Homomorphism f ⊗ g......Page 17
5. Tensor Product of Algebras......Page 19
6. Trilinear Maps......Page 21
7. Exactness......Page 22
8. Homology and Cohomology......Page 28
9. Homotopy......Page 33
10. Direct and Inverse Systems......Page 35
Exercises......Page 40
11. Group Rings......Page 42
12. The Trace Map......Page 44
13. G-Induced Modules......Page 47
14. Complete Resolutions......Page 48
15. The Dependence of Hr(G,M) on the G-Module M......Page 51
16. Existence of Complete Resolutions......Page 56
17. The Uniqueness of Cohomology Groups......Page 61
18. The Computation of H-1(G,M)......Page 65
19. The Standard Complete Resolution......Page 67
20. Extensions......Page 71
21. The Negative Cochain Complex......Page 78
22. Homology and Cohomology of Monoids......Page 84
23. Cohomology of Cyclic Groups......Page 89
24. Trivial Action......Page 94
25. Dimension Subgroups......Page 97
26. The Additive Group of Rationals mod 1......Page 104
27. Polynomial Maps......Page 106
28. Polynomial 2-Cocycles......Page 109
29. Proof of D2(G)=G2......Page 112
30. Remarks......Page 115
31. Induced Homomorphisms......Page 117
32. The Restriction and Corestriction (or Transfer) Maps......Page 125
33. Sylow Subgroups......Page 129
34. Generalization of G-Induced Modules......Page 130
35. The Inflation Map......Page 132
36. Cohomological Triviality......Page 137
37. Cohomological Equivalence......Page 140
38. Outer Automorphisms of Finite p-Groups......Page 145
39. Cohomological Characterization of Finite p-Nilpotent Groups......Page 149
40. Stable Elements......Page 151
41. Frobenius Groups......Page 153
42. Definition of Cup Product......Page 157
43. Properties of Cup Products......Page 161
44. Uniqueness......Page 162
45. Further Properties of Cup Products......Page 165
46. Pairings......Page 166
47. Duality......Page 168
48. Periodicity......Page 173
49. Differential Groups......Page 177
50. How Spectral Sequences Are Used......Page 185
51. The Cartan-Leray Spectral Sequence......Page 188
52. The Spectral Sequence of a Group Extension......Page 191
53. The Hochschild-Serre Spectral Sequence......Page 193
54. Exact Sequences Involving the Terms of Spectral Sequences......Page 199
55. A Decomposition of Hp(G,A) by Means of a Normal Hall Subgroup of G......Page 200
56. The Five-Term Exact Sequence of Hochschild and Serre......Page 202
57. Homology Spectral Sequences......Page 204
58. A Homological Condition for Isomorphism of Nilpotent Groups......Page 206
59. Existence of a Normal p-Complement......Page 209
60. A Condition for the Existence of Free Subgroups......Page 211
61. Presentations and Free Subgroups......Page 213
62. The Schur Multiplicator......Page 216
63. Galois Groups......Page 224
64. Abelian Galois Groups......Page 232
65. Class Formations......Page 237
Bibliography......Page 241
Notation Index......Page 246
Subject Index......Page 247