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

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Brauer maps kill nontrivial idempotent orbit sums

Statement

If $Q ⊴ P$ are $p$ -subgroups and $O$ is a non-singleton $P$ -orbit of primitive central idempotents of $k C_{G} (Q)$ , then ${Br}_{P / Q} (\sum_{f \in O} f) = 0$ . The same coefficient assertion holds for any finite non-singleton orbit of elements of that algebra.

Facts & Assumptions

Given: The stated normal subgroups and non-singleton orbit.

[F1]

The relative Brauer map retains $C_{G} (P)$ coefficients of a $P$ -fixed input. (Relative Brauer homomorphism)

[F2]

A nontrivial orbit of a $p$ -group has size divisible by $p$ . (If a finite $p$ -group $P$ acts on a finite set $X$ , then $∣ X ∣ \equiv ∣ X^{P} ∣ (m o d p)$ )

Proof

technique · direct

1.1

The orbit sum is $P$ -fixed because conjugation permutes its summands, so it is in the relative-map domain. At $x \in C_{G} (P)$ , every conjugate of a fixed representative has the same coefficient: conjugation fixes the basis element $x$ .

2.1

The coefficient of $x$ in the sum is therefore $∣ O ∣$ times that coefficient, which is zero in characteristic $p$ by [F2]. All retained coefficients vanish, proving the assertion. The calculation never used idempotence, establishing the additional finite-orbit clause.

F1F2step 1.1∎

Sources

Jacobsen, Block fusion systems and the center of the group ring, §§1.1 and 2.2, pp.3–8 and 13–18. Local argument and conventions as displayed above.

Depends on

Used by

Maximal Brauer pairs exist and are conjugate Theorem
Unique normal subpair below a Brauer pair Theorem

Dependency tree · two levels

12 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

Jacobsen, Block fusion systems and the center of the group ring, §§1.1 and 2.2, pp.3–8 and 13–18 (standard reference, not scraped)