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

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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Brauer kernel and relative trace support

Statement

For a $p$ -subgroup $P$ , $\ker {Br}_{P} = \sum_{Q < P} {Tr}_{Q}^{P} ((k G)^{Q})$ . If $H \leq G$ , $a \in (k G)^{H}$ , and ${Br}_{P} ({Tr}_{H}^{G} (a)) \neq 0$ , then $P$ is conjugate into $H$ .

Facts & Assumptions

Given: Finite $G$ , characteristic- $p$ field, and the subgroups and fixed elements stated.

[F1]

Brauer truncation retains the $P$ -fixed group basis elements. (Brauer homomorphism for a p subgroup)

[F2]

The fixed-algebra projection is multiplicative. (Brauer homomorphism is multiplicative)

[F3]

Non-singleton $P$ -orbits have cardinality 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 conjugation-fixed space has a basis of orbit sums. A nontrivial orbit with representative $x$ and stabilizer $Q = C_{P} (x) < P$ has sum ${Tr}_{Q}^{P} (x)$ and maps to zero. Conversely for $Q < P$ and $a \in (k G)^{Q}$ , the coefficient of $z \in C_{G} (P)$ in ${Tr}_{Q}^{P} (a)$ is $[P : Q] a_{z} = 0$ . Hence the traces are in the kernel and span precisely the orbit sums removed by truncation. For $P = 1$ the kernel and empty trace sum are both zero.

F1F2F3

2.1

In ${Tr}_{H}^{G} (a)$ , index the conjugates by $G / H$ and group them into $P$ -orbits. At $z \in C_{G} (P)$ , all coefficients within one orbit agree, since conjugation by $P$ fixes $z$ . Every non-singleton orbit contributes zero. A surviving coefficient therefore requires a fixed coset $g H$ , and $P g H = g H$ means $g^{- 1} P g \leq H$ . This proves the trace-support assertion.

F1F3∎

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
Defect groups are maximal Brauer support 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

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)