Representations of Finite Groups

Representations of FInite Groups......Page 1
Contents......Page 4
Preface to the English Edition......Page 7
Preface......Page 8
Acknowledgments......Page 11
1. Definitions and Notations......Page 12
2. Noetherian and Artinian Modules......Page 19
3. The Radical of a Ring......Page 22
4. Idempotents......Page 27
5. Endomorphism Rings......Page 34
6. The Krull-Schmidt-Azumaya Theorem......Page 38
7. Completely Reducible Modules......Page 40
8. Artinian Rings......Page 42
9. Hom and ⊗......Page 53
10. Projective and Injective Modules......Page 63
11. Change of Rings......Page 73
12. Existence of Injective Hulls......Page 80
13. Discrete Valuation Rings......Page 84
14. Algebras over Complete Discrete Valuation Rings......Page 103
Problems......Page 108
1. Fundamental Concepts of Representations......Page 112
2. Algebras over Fields......Page 120
3. Absolutely Irreducible Representations......Page 123
4. Simple Algebras......Page 131
5. Separable Algebras......Page 139
6. The Schur Index......Page 143
7. Crossed Products......Page 148
8. Frobenius Algebras and Symmetric Algebras......Page 156
Problems......Page 174
3. Representations of Groups......Page 178
1. Representations of Groups and Group Rings......Page 179
2. Ordinary Representations......Page 196
3. The Clifford Theory......Page 213
4. Some Brauer Theorems......Page 217
5. Projective Representations......Page 223
6. Introduction to Modular Representation Theory......Page 241
Problems......Page 265
1. Trace Maps......Page 270
2. H-Projective Modules......Page 275
3. Vertices and Sources......Page 280
4. The Green Correspondence......Page 284
5. Green Correspondence and Endomorphism Rings......Page 291
6. Endomorphism Rings of Induced Modules......Page 296
7. The Green Indecomposability Theorem and Its Applications......Page 301
8. Scott Modules......Page 305
Problems......Page 311
5. Theory of Blocks......Page 316
1. Defect Groups of a Block......Page 317
2. The Brauer Homomorphism and the First Main Theorem......Page 323
3. The Brauer Correspondence......Page 330
4. Generalized Decomposition Numbers and the Second Main Theorem......Page 337
5. Blocks and Normal Subgroups......Page 347
6. The Third Main Theorem......Page 360
7. The Clifford Theory of Blocks (The Stable Case)......Page 362
8. Blocks of Factor Groups......Page 369
9. Subpairs and Subsections......Page 377
10. RG as an R[G x G]-Module......Page 382
11. Lower Defect Groups......Page 389
12. The Glauberman Correspondence......Page 396
Problems......Page 398
Solutions to Problems......Page 404
References......Page 418
Postscript......Page 424
List of Notations......Page 426
Index......Page 430