Alphabeta MathMonomial Characters and M Groups
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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
- 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
Representation Theory of Groups
2 · Logical flowchart
3 · Items
Definition
Monomial representations, monomial characters, and M-groups
Lemma
A coset basis makes a monomial representation monomial matrices
Lemma
A faithful irreducible with a noncentral abelian normal subgroup is induced from a proper inertia group
Lemma
A nonabelian supersolvable group has a noncentral abelian normal subgroup, also in its nonabelian quotients
Lemma
Induction commutes with inflation along a normal subgroup
Theorem
Finite supersolvable groups are M-groups
Theorem
Virtual characters are integrally generated by monomial characters from elementary subgroups
Lemma
Kernel of an induced representation is the intersection of the conjugates of the kernel of the inducing representation
Corollary
M-groups are solvable (Taketa)
Remark
Implications and limits for M-groups
Sources
- Tammo tom Dieck, Representation Theory — §4.3, printed pp. 57–58, and §4.6, printed pp. 64–65
- Wen-Wei Li, Yanqi Lake Lectures on Algebra I — Definition 12.5.1 and Theorem 12.5.6, printed pp. 146–148
- SLMath, Character Theory of Finite Groups, Chapter 9 — slides 375–378
- Tammo tom Dieck, Representation Theory — §4.3, printed pp. 57–58
- Wen-Wei Li, Yanqi Lake Lectures on Algebra I — Definition 12.5.1 and the discussion of monomial matrices, printed p. 146
- Tammo tom Dieck, Representation Theory — Proposition 4.3.2, printed pp. 57–58
- Wen-Wei Li, Yanqi Lake Lectures on Algebra I — Lemma 12.5.2 and Lemma 12.5.3, printed pp. 146–147
- Tammo tom Dieck, Representation Theory — Lemma 4.3.3, printed p. 58
- Wen-Wei Li, Yanqi Lake Lectures on Algebra I — Lemma 12.5.2, printed p. 146
- Tammo tom Dieck, Representation Theory — Lemma 4.3.4, printed p. 58
- Wen-Wei Li, Yanqi Lake Lectures on Algebra I — Exercise 12.5.4, printed p. 147
- Tammo tom Dieck, Representation Theory — Theorem 4.3.1 with Lemmas 4.3.3–4.3.4, printed pp. 57–58
- Wen-Wei Li, Yanqi Lake Lectures on Algebra I — Theorem 12.5.6 and its proof, printed pp. 146–148
- Wen-Wei Li, Yanqi Lake Lectures on Algebra I — Theorem 14.3.1 and Corollary 14.3.2, printed pp. 162–164
- Tammo tom Dieck, Representation Theory — Theorem 4.6.3 (monomial induction) with §4.6.2, printed pp. 64–65
- SLMath, Character Theory of Finite Groups, Chapter 9 — slides 387–390 (induced-character kernel lemma $\ker(\alpha^G)\subseteq H$ with proof)
- I. M. Isaacs, Character Theory of Finite Groups — Chapter 5, kernel of an induced character, p. 67 (standard sharp form, located through the book's index)
- SLMath, Character Theory of Finite Groups, Chapter 9 — slides 391–404 (Taketa's theorem, ordered character-degree induction)
- Tammo tom Dieck, Representation Theory — §4.3, Problem 1 (binary tetrahedral group), printed pp. 58–59
- SLMath, Character Theory of Finite Groups, Chapter 9 — slides 391–404 and 408 (Taketa; supersolvable is strictly stronger)