Alphabeta MathBrauers Second Main Theorem
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Characters and the Orthogonality Relations
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Averaging and Character-Theory Prerequisites
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Induced Representations, Frobenius Reciprocity and Applications
- Maschke's Theorem, Complete Reducibility and the Structure of k[G]
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Tensor Products of Modules
- The Group Algebra and Representations of Finite Groups
Commutative Algebra
Group Theory
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Number Theory
Real Analysis
Representation Theory of Groups
2 · Logical flowchart
3 · Items
Lemma
Every finite-group element has unique commuting p- and p-prime parts
Definition
The p-section of a p-element
Definition
Generalized decomposition numbers
Theorem
Generalized decomposition numbers exist and are unique
Definition
Relative projectivity and vertices for integral group lattices
Theorem
Krull-Schmidt holds for finite-rank OH-lattices
Lemma
Integral Mackey decomposition and Higman's criterion for group lattices
Theorem
Green indecomposability for index-p integral induction
Lemma
Central p-subgroups lie in every block defect group
Definition
Brauer subsections and B-subsections
Lemma
Relative projectivity forces character vanishing off the controlling p-section
Theorem
Nagao decomposition for restriction to a centralizer
Lemma
Nagao error terms have zero trace on the relevant p-section
Lemma
Local block projection controls p-section character support
Theorem
Brauer's Second Main Theorem
Corollary
Generalized decomposition columns have corresponding block support
Sources
- Aschbacher–Kessar–Oliver, Fusion Systems in Algebra and Topology, Part IV section 4.4, p. 272
- Craven, The Brauer Correspondence, Chapter 1 section 1.5, pp. 13–14
- Meierfrankenfeld, MTH 912 Class Notes, Definition 6.7.5 and Lemma 6.7.6, pp. 167–168
- Aschbacher–Kessar–Oliver, Fusion Systems in Algebra and Topology, Part IV after Theorem 5.4, pp. 277–278
- Craven, The Brauer Correspondence, section 1.5 and construction preceding Theorem 1.19, pp. 13–14
- Aschbacher–Kessar–Oliver, Fusion Systems in Algebra and Topology, Part IV, pp. 277–278
- Craven, The Brauer Correspondence, sections 2.1–2.2, pp. 19–22
- Aschbacher–Kessar–Oliver, Fusion Systems in Algebra and Topology, Definition 4.1 and Theorem 5.4 proof, pp. 264 and 276–277
- Aschbacher–Kessar–Oliver, Fusion Systems in Algebra and Topology, Part IV, Propositions 1.2–1.9 and 4.9, pp. 228–230 and 267
- Craven, The Brauer Correspondence, sections 2.1–2.2, pp. 18–22
- Craven, The Brauer Correspondence, Proposition 2.3 and sections 2.1–2.2, pp. 19–22; Proposition 3.11, p. 36
- Gyujin Oh, Basic Modular Representation Theory, Theorem 4.1 and proof, pp. 6–7
- Craven, The Brauer Correspondence, Theorem 2.2 and its use in Theorem 2.20, pp. 19 and 28–29
- Craven, The Brauer Correspondence, Lemma 1.2, p. 2
- Meierfrankenfeld, MTH 912 Class Notes, sections 6.6–6.7, pp. 156–165
- Craven, The Brauer Correspondence, section 1.5 and Theorem 1.19, pp. 13–16
- Aschbacher–Kessar–Oliver, Fusion Systems in Algebra and Topology, Theorems 5.4–5.5, pp. 276–278
- Craven, The Brauer Correspondence, Theorem 2.20 and proof, pp. 28–29
- Aschbacher–Kessar–Oliver, Fusion Systems in Algebra and Topology, proof of Theorem 5.4, pp. 276–277
- Craven, The Brauer Correspondence, Theorems 2.17–2.18 and sections 3.1–3.3, pp. 26–36
- Craven, The Brauer Correspondence, Lemma 2.21 and proof, p. 29
- Craven, The Brauer Correspondence, Lemma 2.21 and Theorem 2.22, pp. 29–30
- Meierfrankenfeld, MTH 912 Class Notes, Lemmas 6.7.8–6.7.14, pp. 164–169
- Craven, The Brauer Correspondence, Theorems 1.19 and 2.22, pp. 14 and 29–30
- Meierfrankenfeld, MTH 912 Class Notes, Theorem 6.7.15 and Corollary 6.7.16, pp. 169–171
- Craven, The Brauer Correspondence, Theorem 2.22, pp. 29–30
- Meierfrankenfeld, MTH 912 Class Notes, Theorem 6.7.15, pp. 169–171