The Solution of the Kgv Problem

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ISBN: 9781860949708 | 1860949703
Cover: Hardcover
Copyright: 12/21/2007

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Preface	p. vii
Conjugacy Classes, Characters, and Clifford Theory	p. 1
Class Functions and Characters	p. 1
Induced and Tensor-induced Modules	p. 3
Schur's Lemma	p. 4
Brauer's Permutation Lemma	p. 6
Algebraic Conjugacy	p. 7
Coprime Actions	p. 9
Invariant and Good Conjugacy Classes	p. 10
Nonstable Clifford Theory	p. 12
Stable Clifford Theory	p. 13
Good Conjugacy Classes and Extendible Characters	p. 18
Blocks of Characters and Brauer's k(B) Problem	p. 19
Modular Decomposition and Brauer Characters	p. 19
Cartan Invariants and Blocks	p. 21
Defect and Defect Groups	p. 23
The Brauer-Feit Theorem	p. 25
Higher Decomposition Numbers, Subsections	p. 26
Blocks of p-Solvable Groups	p. 28
Coprime F[subscript p]X-Modules	p. 31
The k(GV) Problem	p. 32
Preliminaries	p. 32
Transitive Linear Groups	p. 34
Subsections and Point Stabilizers	p. 36
Abelian Point Stabilizers	p. 41
Symplectic and Orthogonal Modules	p. 45
Self-dual Modules	p. 45
Extraspecial Groups	p. 47
Holomorphs	p. 49
Good Conjugacy Classes Once Again	p. 54
Some Weil Characters	p. 56
Symplectic and Orthogonal Modules	p. 60
Real Vectors	p. 63
Regular, Abelian and Real Vectors	p. 63
The Robinson-Thompson Theorem	p. 66
Search for Real Vectors	p. 68
Clifford Reduction	p. 71
Reduced Pairs	p. 74
Counting Methods	p. 74
Two Examples	p. 77
Reduced Pairs of Extraspecial Type	p. 82
Nonreal Reduced Pairs	p. 82
Fixed Point Ratios	p. 84
Point Stabilizers of Exponent 2	p. 86
Characteristic 2	p. 90
Extraspecial 3-Groups	p. 92
Extraspecial 2-Groups of Small Order	p. 96
The Remaining Cases	p. 103
Reduced Pairs of Quasisimple Type	p. 110
Nonreal Reduced Pairs	p. 110
Regular Orbits	p. 112
Covering Numbers, Projective Marks	p. 115
Sporadic Groups	p. 119
Alternating Groups	p. 121
Linear Groups	p. 125
Symplectic Groups	p. 129
Unitary Groups	p. 136
Orthogonal Groups	p. 145
Exceptional Groups	p. 147
Modules without Real Vectors	p. 148
Some Fixed Point Ratios	p. 148
Tensor Induction of Reduced Pairs	p. 149
Tensor Products of Reduced Pairs	p. 155
The Riese-Schmid Theorem	p. 156
Nonreal Induced Pairs, Wreath Products	p. 160
Class Numbers of Permutation Groups	p. 170
The Partition Function	p. 170
Preparatory Results	p. 171
The Liebeck-Pyber Theorem	p. 172
Improvements	p. 174
The Final Stages of the Proof	p. 180
Class Numbers for Nonreal Reduced Pairs	p. 180
Counting Invariant Conjugacy Classes	p. 182
Nonreal Induced Pairs	p. 185
Characteristic 5	p. 186
Summary	p. 194
Possibilities for k(GV) = [vertical bar]V[vertical bar]	p. 195
Preliminaries	p. 195
Some Congruences	p. 197
Reduced Pairs	p. 199
Some Consequences for Block Theory	p. 202
Brauer Correspondence	p. 202
Clifford Theory of Blocks	p. 203
Blocks with Normal Defect Groups	p. 207
The Non-Coprime Situation	p. 209
Cohomology of Finite Groups	p. 213
Some Parabolic Subgroups	p. 217
Weil Characters	p. 221
Bibliography	p. 225
List of Symbols	p. 230
Index	p. 231
Table of Contents provided by Ingram. All Rights Reserved.

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