Alphabeta MathBrauers First Main Theorem — Examples
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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
Example
Brauer correspondence for a defect-D8 block of S7 in characteristic 2
Example
Brauer correspondence for SL2(Fp) in defining characteristic
Counterexample
Centralizer containment is sufficient but not necessary
Example
The trivial-defect-group and identity-normalizer boundaries
Sources
- Craven, The Brauer Correspondence, §1.6 example for S7, pp. 13–16
- Saunders, Modular Representation Theory, Examples 4.31 and 5.5
- Saunders, Modular Representation Theory, Example 5.15
- Martínez, Representation Theory of Finite Groups, Theorem 4.10, pp. 27–28
- Farrell–Lassueur, Modular Representation Theory of Finite Groups, Theorem 40.4, §40 (printed pp.8–12 of upload17)