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

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Maximal Brauer pairs detect defect groups

Statement

A $b$ -Brauer pair $(D, e)$ is maximal if and only if $D$ is a defect group of $b$ .

Facts & Assumptions

Given: A $b$ -Brauer pair $(D, e)$ over a field of characteristic $p$ .

[F1]

Every maximal pair has a defect subgroup. (Maximal Brauer pairs exist and are conjugate)

[F2]

Defect groups are exactly maximal nonzero support subgroups. (Defect groups are maximal Brauer support)

[F3]

Pair inclusion entails subgroup inclusion and is equality at equal subgroups. (Brauer pair order is independent of the normal chain)

Proof

technique · direct

1.1

If $(D, e)$ is maximal, [F1] states and proves that $D$ is a defect group. Its hypotheses hold because this is a $b$ -pair.

2.1

Conversely assume $D$ is a defect group and $(D, e) \leq (P, f)$ for a $b$ -pair. Then $D \leq P$ by [F3] and ${Br}_{P} (b) f = f \neq 0$ , so $P$ has nonzero support. Maximality in [F2] forces $P = D$ . Inclusion at equal subgroups in [F3] now forces $f = e$ . Thus no strictly larger pair exists.

F2F3∎

Sources

Jacobsen, Block fusion systems and the center of the group ring, Lemma 2.32 and Theorem 2.33, pp.18–19; general-field lifting proved locally. Local argument and conventions as displayed above.

Depends on

Used by

Dependency tree · two levels

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Sources

Jacobsen, Block fusion systems and the center of the group ring, Lemma 2.32 and Theorem 2.33, pp.18–19; general-field lifting proved locally (standard reference, not scraped)