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A faithful irreducible with a noncentral abelian normal subgroup is induced from a proper inertia group

Statement

Let $G$ be a finite group, let $A ⊴ G$ be an abelian normal subgroup (Normal subgroup: invariance under conjugation) that is not contained in the center $Z (G)$ , and let $V$ be a faithful irreducible finite-dimensional complex representation of $G$ .

Then for every irreducible constituent $λ$ of the restriction of $V$ to $A$ the inertia group $I_{G} (λ)$ is a proper subgroup of $G$ , and there is an irreducible representation $W$ of $I_{G} (λ)$ lying over $λ$ with $V ≅ {Ind}_{I_{G} (λ)}^{G} W .$

Facts & Assumptions

Given: A finite group $G$ , an abelian normal subgroup $A ⊴ G$ with $A ⊈ Z (G)$ , a faithful irreducible finite-dimensional complex representation $ρ : G \to GL (V)$ , and an irreducible constituent $λ$ of the restriction of $V$ to $A$ .

[F1]

Every irreducible representation of a finite abelian group over a splitting field is one-dimensional, and $C$ is a splitting field for every finite group. (Every irreducible representation of a finite abelian group over a splitting field is one-dimensional, A cyclotomic field splits a finite group).

[F2]

For finite $G$ , $N ⊴ G$ , $θ \in Irr (N)$ and $χ \in Irr (G ∣ θ)$ with $I = I_{G} (θ)$ there is a positive integer $e$ with ${Res}_{N}^{G} χ = e \sum_{g I \in G / I}^{g} θ$ , so the constituents of the restriction are exactly the distinct conjugates of $θ$ , each with multiplicity $e$ ; the restriction is isotypical precisely when $I = G$ . (Clifford restriction formula).

[F3]

$^{g} θ (n) = θ (g^{- 1} n g)$ and $I_{G} (θ) = {g \in G :^{g} θ = θ}$ is a subgroup with $N \leq I_{G} (θ) \leq G$ ; $Irr (H ∣ θ)$ consists of the irreducible characters of $H$ whose restriction to $N$ contains $θ$ . (Inertia group and characters lying above a normal type).

[F4]

Induction gives a bijection $Irr (I_{G} (θ) ∣ θ) \to Irr (G ∣ θ)$ , whose inverse takes the $θ$ -isotypical component; in particular every irreducible $G$ -module lying over $θ$ is induced from its $θ$ -isotypical component. (Clifford correspondence).

[F5]

The representation $ρ$ , and with it $V$ , is faithful: $ρ (g) = {id}_{V}$ implies $g = 1$ . (Intertwiners, the spaces ${Hom}_{G} (V, W)$ and ${End}_{G} (V)$ , equivalent representations, and faithful representations).

[F6]

$Z (G) = {z \in G : z g = g z for all g \in G}$ . (The center $Z (G)$ of a group).

[A1]

$ρ$ is a homomorphism into $GL (V)$ , so $ρ (g^{- 1} a g) = ρ (g)^{- 1} ρ (a) ρ (g)$ .

Proof

technique · direct

1.1

Restricting $ρ$ to the abelian normal subgroup $A$ gives a representation of $A$ whose irreducible constituents are one-dimensional by [F1], because $C$ is a splitting field for $A$ ; thus the constituent $λ$ of the restriction is a group homomorphism $λ : A \to C^{\times}$ .

F1given

2.1

Apply [F2] with $N = A$ and $χ = χ_{V}$ : the constituents of the restriction of $V$ to $A$ are exactly the distinct conjugates $^{g} λ$ with $g \in G$ , each occurring with one positive multiplicity $e$ , and by [F3] the conjugate $^{g} λ$ equals $λ$ precisely when $g \in I_{G} (λ)$ . Hence the restriction of $V$ to $A$ is $λ$ -isotypical, that is all its constituents equal $λ$ , precisely when $I_{G} (λ) = G$ .

F2F3step 1.1

3.1

Suppose now, for the sake of a contradiction, that $I_{G} (λ) = G$ . By step 2.1 the only constituent of the restriction of $V$ to $A$ is $λ$ , so its character is a positive multiple of $λ$ ; equivalently $ρ (a) = λ (a) {id}_{V}$ for every $a \in A$ .

step 2.1given

4.1

Let $a \in A$ and $g \in G$ . Using step 3.1 and [A1], $ρ (g^{- 1} a g) = ρ (g)^{- 1} ρ (a) ρ (g) = λ (a) {id}_{V} = ρ (a)$ . Since $ρ$ is faithful by [F5], this gives $g^{- 1} a g = a$ ; as $g$ was arbitrary, $a$ commutes with every element of $G$ , so $a \in Z (G)$ by [F6]. Hence $A \subseteq Z (G)$ , contradicting the hypothesis $A ⊈ Z (G)$ . Therefore $I_{G} (λ) \neq G$ , and since $A \leq I_{G} (λ) \leq G$ by [F3], the inertia group $I_{G} (λ)$ is a proper subgroup of $G$ .

F3F5F6step 3.1given

5.1

Since $V$ is irreducible and $λ$ is a constituent of its restriction to $A$ , the module $V$ lies over $λ$ . By [F4] the $λ$ -isotypical component $W = V_{λ}$ is an irreducible $I_{G} (λ)$ -module lying over $λ$ and induction gives $V ≅ {Ind}_{I_{G} (λ)}^{G} W$ . As $λ$ was an arbitrary constituent of the restriction to $A$ , both assertions hold for every such constituent.

F4step 1.1step 4.1∎

Depends on

Used by

Finite supersolvable groups are M-groups Theorem

Dependency tree · two levels

26 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Tammo tom Dieck, Representation Theory — Proposition 4.3.2, printed pp. 57–58 (standard reference, not scraped)
Wen-Wei Li, Yanqi Lake Lectures on Algebra I — Lemma 12.5.2 and Lemma 12.5.3, printed pp. 146–147 (standard reference, not scraped)