LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The stabilizer of a nonzero isotypical component

Statement

Let $G$ be finite, $N ⊴ G$ , $V$ an irreducible complex $G$ -module, and $θ$ an occurring constituent of $V ∣_{N}$ . The setwise stabilizer of the nonzero component $V_{θ}$ is exactly $I_{G} (θ)$ . Consequently $V_{θ}$ is an $I_{G} (θ)$ -module.

Facts & Assumptions

Given: The groups, modules, characters, and hypotheses in the statement. All representations here are finite-dimensional complex left representations.

[F1]

Translation sends $V_{θ}$ to $V_{^{g} θ}$ , and distinct isotypical components are direct summands. (Translation permutes normal isotypical components).

Proof

technique · direct

1.1

If $g \in I_{G} (θ)$ , then $^{g} θ = θ$ , so translation gives $g V_{θ} = V_{θ}$ . Thus the inertia group preserves the component.

F1given

2.1

If $g V_{θ} = V_{θ}$ , translation gives $V_{^{g} θ} = V_{θ} \neq 0$ . Distinct components have zero intersection, so the two types coincide and $g \in I_{G} (θ)$ . Restricting the action therefore gives the claimed module. The nonzero hypothesis is essential: the zero subspace has all of $G$ as stabilizer.

F1step 1.1given∎

Depends on

Translation permutes normal isotypical components

Used by

Reconstruction from the inertia component Lemma

Dependency tree · two levels

4 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 — §4.2 p.54 before Proposition 4.2.2 (standard reference, not scraped)