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

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The induction subgroup is an ideal

Statement

For every family $F$ of subgroups of a finite group $G$ , $I_{F} (G)$ is an ideal of $R (G)$ .

Facts & Assumptions

[F1]

The cited prerequisite is Induction and restriction satisfy the projection formula on character rings.

Proof

Given: $H \in F$ , $θ \in R (H)$ , and $χ \in R (G)$ .

1.1

The projection formula gives ${Ind}_{H}^{G} θ \cdot χ = {Ind}_{H}^{G} (θ \cdot {Res}_{H}^{G} χ)$ .

F1given

2.1

Its right side is one of the defining summands of $I_{F} (G)$ ; additivity handles finite sums and additive inverses, so multiplication by every $χ$ preserves the subgroup. ∎

step 1.1

Depends on

Used by

Elementary detection at a fixed element Lemma
Brauer induction Theorem

Dependency tree · two levels

8 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

Wen-Wei Li, Yanqi Lake Lectures on Algebra I, Lemma 14.2.6 (standard reference, not scraped)