Modular Representations of Finite Groups (eBook)

Front Cover 1
Modular Representations of Finite Groups 4
Copyright Page 5
Contents 6
Preface 10
Note to the Reader 12
Notation 14
Chapter I. Representation Modules 18
1.1 Group algebras and modules 18
1.2 Reducible and irreducible modules 22
1.3 Semisimple rings and the Wedderburn structure theorem 25
1.4 Tensor products 30
1.5 The number of irreducible KG-modules 32
1.6 Indecomposable modules 37
1.7 Absolutely indecomposable and absolutely irreducible modules 41
1.8 Principal indecomposable modules 43
1.9 Composition factors and intertwining numbers 46
1.10 Notes and comments 49
Chapter II. Induced Modules and Characters 50
2.1 Induced modules 50
2.2 Clifford's theorem 54
2.3 Group characters 56
2.4 The theory of ordinary characters 60
2.5 Induced characters 65
2.6 Brauer's theorem on induced characters 69
2.7 Splitting fields 73
2.8 Notes and comments 77
Chapter III. Modular Representations and Characters 79
3.1 The p-adic integers 79
3.2 p-adic algebras 82
3.3 Ordinary and modular representations 86
3.4 Lifting idempotents 90
3.5 The case where p does not divide |G| 92
3.6 Modular characters 94
3.7 Cartan invariants, decomposition numbers, and orthogonality relations 96
3.8 Modular characters of p-solvable groups 101
3.9 Notes and comments 105
Chapter IV. Blocks of Group Algebras 106
4.1 Blocks 106
4.2 Classifying modules, characters, and idempotents into blocks 110
4.3 Defect groups 117
4.4 Further analysis of the Cartan matrix and decomposition matrix 121
4.5 The characters in a block of given defect 124
4.6 Blocks of small defect 129
4.7 Notes and comments 131
Chapter V. The Theory of Indecomposable Modules 133
5.1 Relatively projective modules 133
5.2 Vertices and sources 137
5.3 Green's theorem 140
5.4 The degrees of indecomposable modules 144
5.5 Vertices and defect groups 146
5.6 Restriction of indecomposable modules 147
5.7 Jordan's theorem in characteristic p 150
5.8 Notes and comments 157
Chapter VI. The Main Theorems of Brauer 159
6.1 The Brauer homomorphism 159
6.2 Blocks with normal p-subgroups 161
6.3 The Brauer correspondence: The First Main Theorem 166
6.4 Extension of the First Main Theorem 169
6.5 Generalized decomposition numbers: The Second Main Theorem 174
6.6 Principal blocks: The Third Main Theorem 179
6.7 The characters in the principal block 181
6.8 Notes and comments 185
Chapter VII. Fusion of 2-Groups 186
7.1 Further results on generalized decomposition numbers 186
7.2 Some technical lemmas 191
7.3 Groups with Sylow 2-subgroups of type (2m, 2m) 195
7.4 Groups with quaternion Sylow 2-subgroups 201
7.5 Glauberman's Z*-theorem 208
7.6 Notes and comments 213
Chapter VIII. Blocks with Cyclic Defect Groups 214
8.1 Extending characters from normal subgroups 214
8.2 Blocks with normal cyclic defect groups 216
8.3 Groups with cyclic Sylow p-subgroups 222
8.4 Some technical lemmas 223
8.5 Groups of order g = pg0, with p X g0 225
8.6 Groups with a faithful representation of degree d < 1/2( p – 1)
8.7 Criteria for normal Sylow p-groups 237
8.8 Notes and comments 243
References 246
Index 256