PropositionStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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A faithful irreducible is induced from a proper inertia subgroup

Statement

Let $G$ be finite and let $A ◃ G$ be abelian and noncentral. Every faithful irreducible complex representation $V$ of $G$ is induced from an irreducible representation of a proper inertia subgroup of $G$ .

Facts & Assumptions

[F1]

The cited prerequisite is If $char k ∤ ∣ G ∣$ , every finite-dimensional representation of $G$ is completely reducible.

Proof

Given: $V$ is faithful and irreducible, and $λ$ is a linear constituent of ${Res}_{A}^{G} V$ .

1.1

Complete reducibility decomposes $V ∣_{A}$ into its linear weight spaces. The translates of the $λ$ -weight space are the weight spaces in its $G$ -orbit, and their direct sum is $V$ by irreducibility.

F1given

2.1

If the inertia group $G_{λ}$ were $G$ , every $a \in A$ would act by a scalar on $V$ ; faithfulness would then make $A$ central, contrary to hypothesis. Thus $G_{λ} < G$ , and the direct sum of its translates identifies $V$ with the induction of its $λ$ -isotypical component. ∎

step 1.1

Depends on

Used by

Finite supersolvable groups are monomial Theorem

Dependency tree · two levels

15 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 (standard reference, not scraped)