Representation Theory of Finite Groups

Chapter I. Foundations 1
1. Introduction 1
2. Group Characters 12
3. Representation Modules 13
4. Application of Ideas and Results from Group Theory 17
5. The Regular Representation 23
Exercises 24
Chapter II. Representation Theory of Rings with Identity 28
6. Some Fundamental Lemmas 28
Exercise 34
7. The Principal Indecomposable Representations 34
8. The Radical of a Ring 37
9. Semisimple Rings 40
10. The Wedderburn Structure Theorems for Semisimple Rings 42
11. Intertwining Numbers 47
12. Multiplicities of the Indecomposable Representation 53
13. The Generalized Burnside Theorem 55
Exercises 56
Chapter III. The Representation Theory of Finite Groups 58
14. The Group Algebra 58
15. The Regular Representation of a Group 60
16. Semisimplicity of the Group Algebra 60
17. The Center of the Group Algebra 64
18. The Number of Inequivalent Irreducible Representations 65
19. Relations on the Irreducible Characters 67
20. The Module of Characters over the Integers 71
21. The Kronecker Product of Two Representations 72
Exercises 75
22. Linear Characters 76
Exercises 77
23. Induced Representations and Induced Characters 77
Exercises 83
Chapter IV. Applications of the Theory of Characters 85
24. Algebraic Numbers 85
25. Some Results from the Theory of Characters 87
26. Normal Subgroups and the Character Table 92
A. The Existence of Normal Subgroups 92
B. The Determmation of All Normal Subgroups 93
27. Some Classical Theorems 94
Exercises 98
Chapter V. The Construction of Irreducible Representations 99
28. Primitive Idempotents 99
29. Some examples of Group Representations 105
1. Cyclic Groups 105
2. Abelian Groups 106
3. The Symmetric Groups S_n 106
Exercises 119
Chapter VI. Modular Representations 120
30. General Remarks 120
31. P-Regular Elements of a Finite Group 122
32. Conditions for Two Representations to Have the Same Composition Factors 125
33. The Brauer Characters 128
34. Integral Representations 130
Exercise 134
35. Ordinary and Modular Representations of Algebras 134
1. Arithmetic in an Algebra 134
Exercise 136
2. Connection with Integral Representations 136
36. P-Adic Fields 137
1. General Definition and Properties 137
2. Ordinary Valuation. Metrical Properties 139
3. Completion of a P-Adic Field 140
4. p-Adic Valuation of the Rational Field 143
5. Extension of the P-Adic Valuation to Algebraic Number Fields 143
37. Algebras over a P-Adic Field 148
1. Notation 148
2. Preliminary Results 149
38. A Connection between the Intertwining Numbers 157
39. Modular Representations of Groups 159
40. Cartan Invariants and Decomposition Numbers 161
41. Character Relations 164
42. Modular Orthogonality Relations 169
Appendix 173
1. Groups 173
2. Rings, Ideals, and Fields 176
BIBLIOGRAPHY 180
SUBJECT INDEX 183