LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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.

Relative Brauer homomorphisms are transitive

Statement

Let $P \leq G$ be a $p$ -subgroup. If $Q ⊴ R ⊴ P$ and also $Q ⊴ P$ , then on $(k C_{G} (Q))^{P}$ one has ${Br}_{P / R} \circ {Br}_{R / Q} = {Br}_{P / Q}$ . Also ${Br}_{P / Q} ({Br}_{Q} (a)) = {Br}_{P} (a)$ for $a \in (k G)^{P}$ when $Q ⊴ P$ .

Facts & Assumptions

Given: The $p$ -subgroup $P$ and the normalities in the statement; normality is not assumed transitive.

[F1]

Relative maps truncate from one centralizer to the smaller centralizer on the appropriate fixed domain. (Relative Brauer homomorphism)

[F2]

Brauer maps commute with conjugation. (Brauer homomorphism is conjugation equivariant)

Proof

technique · direct

1.1F1F2

For $a \in (k C_{G} (Q))^{P}$ , truncating to $C_{G} (R)$ leaves a $P$ -fixed element: $P$ normalizes both centralizers by the stated normalities and permutes their basis coefficients, so [F2] applies. Thus the composite has the displayed domain. Since $C_{G} (P) \subseteq C_{G} (R) \subseteq C_{G} (Q)$ , truncating first to the middle set and then to the smallest retains exactly the same coefficients as truncating directly to the smallest.

2.1F1F2∎

If instead $a \in (k G)^{P}$ and $Q ⊴ P$ , equivariance puts ${Br}_{Q} (a)$ in $(k C_{G} (Q))^{P}$ . Truncation to $C_{G} (Q)$ and then $C_{G} (P)$ retains exactly the $C_{G} (P)$ coefficients of $a$ . This proves the second identity, also when any two subgroups agree.

Sources

Webb, A Course in Finite Group Representation Theory, §§11.3, 11.6 and 12.3–12.5, especially pp.240–245. Local argument and conventions as displayed above.

Depends on

Used by

Brauer pair order is independent of the normal chain Theorem

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

Webb, A Course in Finite Group Representation Theory, §§11.3, 11.6 and 12.3–12.5, especially pp.240–245 (standard reference, not scraped)