Groups (eBook)

Preface 5
Notation 8
Table of Contents 9
1 Introductory Notions 12
1.1 Definitions and First Theorems 12
1.2 Cosets and Lagrange’s Theorem 35
1.3 Automorphisms 45
2 Normal Subgroups, Conjugation and Isomorphism Theorems 49
2.1 Product of Subgroups 49
2.2 Normal Subgroups and Quotient Groups 50
2.3 Conjugation 60
2.4 Normalizers and Centralizers of Subgroups 69
2.5 H¨older’s Program 73
2.6 Direct Products 77
2.7 Semidirect Products 83
2.8 Symmetric and Alternating Groups 88
2.9 The Derived Group 92
3 Group Actions and Permutation Groups 97
3.1 Group actions 97
3.2 The Sylow Theorem 109
3.3 Burnside’s Formula and Permutation Characters 127
3.4 Induced Actions 135
3.5 Permutations Commuting with an Action 138
3.6 Automorphisms of Symmetric Groups 144
3.7 Permutations and Inversions 146
3.8 Some Simple Groups 153
3.8.1 The Simple Group of Order 168 153
3.8.2 Projective Special Linear Groups 157
4 Generators and Relations 164
4.1 Generating Sets 164
4.2 The Frattini Subgroup 169
4.3 Finitely Generated Abelian Groups 173
4.4 Free abelian groups 179
4.5 Projective and Injective Abelian Groups 187
4.6 Characters of Abelian Groups 190
4.7 Free Groups 192
4.8 Relations 197
4.8.1 Relations and simple Groups 201
4.9 Subgroups of Free Groups 203
4.10 The Word Problem 207
4.11 Residual Properties 209
5 Nilpotent Groups and Solvable Groups 214
5.1 Central Series and Nilpotent Groups 214
5.2 p-Nilpotent Groups 232
5.3 Fusion 238
5.4 Fixed-Point-Free Automorphisms and Frobenius Groups 241
5.5 Solvable Groups 246
6 Representations 262
6.1 Definitions and examples 262
6.1.1 Maschke’s Theorem 266
6.2 Characters 268
6.3 The Character Table 284
6.3.1 Burnside’s Theorem and Frobenius Theorem 289
6.3.2 Topological Groups 294
7 Extensions and Cohomology 298
7.1 Crossed Homomorphisms 298
7.2 The First Cohomology Group 301
7.3 The Second Cohomology Group 310
7.3.1 H1 and Extensions 316
7.3.2 H2(p,A) for p Finite Cyclic 317
7.4 The Schur Multiplier 321
7.4.1 Projective Representations 322
7.4.2 Covering Groups 324
7.4.3 M(p) and Presentations of p 329
8 Solution to the exercises 335
8.1 Chapter 1 335
8.2 Chapter 2 337
8.3 Chapter 3 343
8.4 Chapter 4 354
8.5 Chapter 5 359
8.6 Chapter 6 366
References 371
Index 373