Alphabeta MathBrauers Second 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
p-sections and Brauer subsections in S3
Example
The Second Main Theorem at u=1 is block-diagonal decomposition
Example
A p-section with no local block inducing to the chosen global block
Sources
- Aschbacher–Kessar–Oliver, Fusion Systems in Algebra and Topology, Example 4.25 and Theorems 5.1 and 5.4, pp. 274–277
- Craven, The Brauer Correspondence, sections 1.5–1.6, pp. 13–16
- Meierfrankenfeld, MTH 912 Class Notes, Theorem 6.7.15 and Corollary 6.7.16, pp. 169–171
- Craven, The Brauer Correspondence, section 1.5, pp. 13–14
- Aschbacher–Kessar–Oliver, Fusion Systems in Algebra and Topology, Example 4.25 and Theorem 5.4, pp. 274–277
- Meierfrankenfeld, MTH 912 Class Notes, Theorem 6.7.15, pp. 169–171