Alphabeta MathBrauers First 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
- 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
Representation Theory of Groups
2 · Logical flowchart
3 · Items
Theorem
Modular block central characters correspond to blocks
Definition
A block induced from a subgroup
Lemma
Induced blocks have controlled defect
Lemma
Centralizer containment makes block induction well-defined
Lemma
Block induction is transitive when both stages are defined
Lemma
Normal p-subgroups fix block idempotents under Brauer projection
Lemma
A global block of defect D determines a local block of defect D
Lemma
Every local full-defect block induces to a global block of defect D
Lemma
Distinct local full-defect blocks induce to distinct global blocks
Theorem
Brauer's First Main Theorem
Definition
The Brauer correspondent of a block
Theorem
Corresponding block bimodules are Green correspondents
Lemma
Tensoring preserves relative projectivity for finite-group modules
Theorem
Brauer–Green block compatibility
Corollary
Every block contains a module whose vertex is a full defect group
Sources
- Martínez, Representation Theory of Finite Groups, Theorem 2.11, p. 15
- Craven, The Brauer Correspondence, central-character setup, pp. 4–6
- Saunders, Modular Representation Theory, Definition 5.13
- Farrell–Lassueur, Modular Representation Theory of Finite Groups, Definition 40.1, §40 (printed pp.8–12 of upload17)
- Saunders, Modular Representation Theory, Lemma 5.14(i)
- Farrell–Lassueur, Modular Representation Theory of Finite Groups, Proposition 40.3(i), §40 (printed pp.8–12 of upload17)
- Farrell–Lassueur, Modular Representation Theory of Finite Groups, Proposition 40.3(iii), §40 (printed pp.8–12 of upload17)
- Martínez, Representation Theory of Finite Groups, Theorem 4.5, pp. 24–25
- Saunders, Modular Representation Theory, Lemma 5.14(ii)
- Farrell–Lassueur, Modular Representation Theory of Finite Groups, Proposition 40.3(ii), §40 (printed pp.8–12 of upload17)
- Martínez, Theorem4.5 proof, normal-p orbit localization; local central-idempotent consequence
- Farrell–Lassueur, Modular Representation Theory of Finite Groups, Theorem 40.4, §40 (printed pp.8–12 of upload17)
- Craven, The Brauer Correspondence, Theorem 1.12, pp. 9–10
- Saunders, Modular Representation Theory, Theorem 5.16
- Martínez, Representation Theory of Finite Groups, Theorem 4.10, pp. 27–28
- Craven, The Brauer Correspondence, Definition following Theorem 1.12, p. 10
- Farrell–Lassueur, Modular Representation Theory of Finite Groups, Exercise 31(a) as used in Theorem 40.5, §40 (printed pp.8–12 of upload17)
- Farrell–Lassueur, Modular Representation Theory of Finite Groups, Theorem 40.5, §40 (printed pp.8–12 of upload17)
- Saunders, Modular Representation Theory, Theorem 5.17 and Corollary 5.18
- Farrell–Lassueur, Modular Representation Theory of Finite Groups, Corollary 40.7, §40 (printed pp.8–12 of upload17)
- Saunders, Modular Representation Theory, Corollary 5.19