Cohomology of Groups (eBook)

Front Cover 1
Cohomology of Groups 4
Copyright Page 5
Contents 10
Preface 8
Chapter I. Cohomology Groups of G in A 14
1-1. Differential Groups 14
1-2. Herbrand’s Lemma 21
1-3. G-Complexes (Positive Part) 25
1-4. G-Complexes (Negative Part) 31
1-5. Cohomology Groups of G in A 39
1-6. Problems and Supplements 48
Chapter II . Mappings of Cohomology Groups 58
2-1. Homomorphism of Pairs 58
2-2. Independence of the Complex 68
2-3. Conjugation, Restriction, and Inflation 77
2-4. The Transfer or Corestriction 83
2-5. Explicit Formulas 90
Chapter III . Some Properties of Cohomology Groups 100
3-1. Miscellaneous Facts 100
3-2. Cyclic Groups and the Herbrand Quotient 106
3-3. Dimension Shifting 114
3-4. The Inflation-Restriction Sequence 121
3-5. The Group H–2(G, Z) 125
3-6. Cohomological Equivalence 132
3-7. Problems and Supplements 138
Chapter IV. The Cup Product 149
4-1. Cup Products 149
4-2. Properties of the Cup Product 158
4-3. Cup Products for a Pairing 168
4-4. The Duality Theorem 176
4-5. The Nakayama Map 183
4-6. Problems and Supplements 194
Chapter V. Group Extensions 208
5-1. The Extension Problem and H2 208
5-2. Commutator Subgroups in Group Extensions 219
5-3. Factor Extensions 226
5-4. The Principal Ideal Theorem 231
Chapter VI. Abstract Class Field Theory 239
6- 1. Formations 239
6-2. Field Formations 245
6-3. Class Formations 250
6-4. The Main Theorem 256
6-5. Reciprocity Law and Norm Residue Symbol 262
6-6. The Existence Theorem 275
References 280
Subject Index 284