Alphabeta MathBlocks Defect Groups and the Brauer Homomorphism
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- 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
- 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
- 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
Number Theory
Real Analysis
2 · Logical flowchart
3 · Items
Definition
Block bimodule for the double group
Lemma
Group algebra bimodule is induced from the diagonal
Lemma
Relative projectivity mackey intersections for finite modules
Theorem
Block bimodule has a diagonal vertex
Definition
Defect group and numerical defect of a block
Theorem
Defect groups of a block are conjugate
Lemma
Block relative trace characterizes diagonal projectivity
Theorem
Vertices of modules in a block lie in a defect group
Proposition
Principal block has sylow defect
Lemma
A projective simple in a symmetric block forces a matrix block
Theorem
Defect zero blocks are simple algebras
Definition
Brauer homomorphism for a p subgroup
Theorem
Brauer homomorphism is multiplicative
Lemma
Brauer homomorphism is conjugation equivariant
Definition
Relative Brauer homomorphism
Lemma
Relative Brauer homomorphisms are transitive
Proposition
Central idempotents under the Brauer homomorphism
Lemma
Brauer kernel and relative trace support
Lemma
Block centre locality and trace ideal sums
Theorem
Defect groups are maximal Brauer support
Lemma
Finite-dimensional algebras admit primitive idempotent decompositions
Definition
Brauer pair for a block
Definition
Normal inclusion of Brauer pairs
Lemma
Brauer maps kill nontrivial idempotent orbit sums
Theorem
Unique normal subpair below a Brauer pair
Lemma
Brauer images retain the surviving primitive idempotents
Theorem
Brauer pair order is independent of the normal chain
Theorem
Every Brauer pair determines a unique global block
Lemma
Finite-dimensional commutative algebras decompose into local factors
Lemma
Idempotents lift through finite commutative algebra quotients
Theorem
Maximal Brauer pairs exist and are conjugate
Theorem
Maximal Brauer pairs detect defect groups
Lemma
Restriction to a containing p subgroup retains a vertex
Lemma
Transitive p-group permutation modules have point-stabilizer vertices
Lemma
Block defect is an intersection of two sylow subgroups
Corollary
Normal p core lies in every block defect group
Corollary
Block defect groups are p radical
Sources
- Webb, A Course in Finite Group Representation Theory, §§5.2, 11.3, 11.6, 12.3–12.5; especially Lemma 12.4.4 and Theorem 12.4.5, pp.240–241
- Webb, A Course in Finite Group Representation Theory, §§11.3, 11.6 and 12.3–12.5, especially pp.240–245
- Jacobsen, Block fusion systems and the center of the group ring, §§1.1 and 2.2, pp.3–8 and 13–18
- AKO, Fusion Systems in Algebra and Topology, IV §1 Proposition 1.4 and IV §2 Lemma 2.8; arbitrary-field polynomial proof supplied here
- AKO, Fusion Systems in Algebra and Topology, IV §2 Theorem 2.10, Lemmas 2.11–2.12 and Proposition 2.14, printed pp.180–183; six-clause formulation retained from BKY Theorem 2.2
- Jacobsen, Block fusion systems and the center of the group ring, Lemma 2.32 and Theorem 2.33, pp.18–19; general-field lifting proved locally