Alphabeta MathFrobenius Groups and the Normal Complement 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
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- 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
- Semidirect Products, Automorphism Groups and Split Extensions
- 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
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
2 · Logical flowchart
3 · Items
Definition
Frobenius complement and frobenius group
Proposition
Frobenius permutation action characterization
Definition
Frobenius kernel set
Lemma
Frobenius kernel cardinality
Definition
Induced class functions and restricted class functions
Lemma
Frobenius reciprocity for class functions
Lemma
Zero at identity induction restriction for a frobenius complement
Lemma
Frobenius character extension construction
Lemma
Frobenius character extension is irreducible
Lemma
Frobenius kernel is an intersection of character kernels
Theorem
Frobenius kernel theorem
Corollary
Frobenius semidirect product decomposition
Proposition
Frobenius groups and fixed point free actions
Definition
The p-prime core of a finite group
Definition
Normal p complement and p nilpotent group
Definition
Transfer homomorphism for a finite index subgroup
Lemma
Transfer is independent of the transversal
Lemma
Transfer is a homomorphism
Lemma
Transfer cycle decomposition formula
Proposition
Equivalent forms of having a normal p complement
Definition
P residual of a finite group
Lemma
Sylow subgroups of a normal subgroup are intersections with Sylow subgroups
Lemma
P residual is generated by p prime elements and idempotent
Lemma
Abelian sylow fusion in its normalizer
Theorem
Burnside normal p complement theorem
Definition
P local normalizer for normal complement theory
Definition
Control of fusion in a sylow p subgroup
Lemma
Proper subgroup of a finite p group is properly normalized local
Lemma
Normal p complements pass to subgroups and p local normalizers
Lemma
Fusion control and centralizer transitivity are equivalent
Lemma
Local sylow conjugacy ascent for fusion
Lemma
Local normal p complements force control of fusion
Lemma
Normal p subgroup has proper commutator in a p group
Lemma
Fusion control forces trivial Sylow intersection with the p residual
Theorem
Frobenius normal p complement theorem
Lemma
Sylow times normal subgroup covers when the index is a p-power
Lemma
The local automizer condition gives centralizer conjugacy of Sylow subgroups
Lemma
P automizer condition implies fusion control
Corollary
Frobenius automizer criterion for p nilpotence
Sources
- Alex Bartel, Introduction to Representation Theory of Finite Groups, §6.1
- Hans Kurzweil and Bernd Stellmacher, The Theory of Finite Groups, §§7.1–7.2
- Peter Webb, A Course in Finite Group Representation Theory, §4.3
- Paul Flavell, An Introduction to Transfer and Fusion in Finite Groups, §§2–5
- David Craven, Finite Group Theory, Lecture 3