Modular Representations of Finite Groups

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Introduction

Chapter 1: Semisimple Modules

1.1 Semisimple Rings
1.2 Simple Modules and $p$-regular classes

Chapter 2: Projective Modules

2.1 Projective Modules
2.2 Indecomposable Modules
2.3 Duality

Chapter 3: Subgroups

3.1 Induced Modules
3.2 Vertices and Sources
3.3 Trivial Intersections
3.4 Green correspondence for $SL(2,p)$

Chapter 4: Lift to Characteristic 0

4.1 Projective Envelopes
4.2 Lifting projective modules
4.3 The cde triangle
4.4 Decomposition Matrices

Chapter 5: Brauer Characters

5.1 Splitting Fields
5.2 Choosing a prime
5.3 Brauer Characters
5.4 Orthogonality for Brauer Characters

Chapter 6: Blocks

6.1 Blocks
6.2 The Center of $k[G]$
6.3 Block orthogonality
6.4 The Brauer correspondence (cyclic Sylow)
6.5 The defect group of a block

Chapter 7: Examples

7.1 Cyclic group of order $p$
7.2 $S_3$, $p=3$
7.3 Dihedral Group of Order 12
7.4 $SL(2,3)$
7.5 $SL(2,5)$
7.6 $SL(2,7)$
7.7 $GL(2,3)$
7.8 $SL(3,2)$
7.9 $SL(3,3)$
7.10 $A_5$

Chapter 8: Brauer's Theorem and applications

8.1 Brauer's Theorem
8.2 Application: $GL(2)$
8.3 Relative Brauer's Theorem
8.4 Properties of the cde triangle