Representation Theory of Finite Groups

Description

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