TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-12

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Every Brauer pair determines a unique global block

Statement

Every local Brauer pair $(P, e)$ belongs to exactly one global block $b$ of $k G$ . Conjugate local pairs belong to the same global block, and comparable local pairs belong to the same global block.

Facts & Assumptions

Given: A finite group $G$ , a field $k$ of characteristic $p$ , and local Brauer pairs.

[F1]

Every subgroup of a pair has a unique subpair; at a normal subgroup inclusion is normal inclusion. (Brauer pair order is independent of the normal chain)

[F2]

Conjugation commutes with Brauer projection. (Brauer homomorphism is conjugation equivariant)

Proof

technique · direct

1.1

Apply unique descent to $1 \leq P$ . Its unique pair $(1, b)$ has $b$ a block of $k C_{G} (1) = k G$ . Since $1 ⊴ P$ , the normal criterion says exactly that $b$ is $P$ -fixed and ${Br}_{P} (b) e = e$ . A global block is central, hence $P$ -fixed. Thus descent is equivalent to membership, proving existence and uniqueness of the global block.

2.1

For $g \in G$ , centrality gives $^{g} b = b$ ; equivariance gives ${Br}_{^{g} P} (b)^{g} e =^{g} ({Br}_{P} (b) e) =^{g} e$ . Therefore conjugate pairs have the same block by step 1.1. If $(Q, f) \leq (P, e)$ , the unique descent $(1, c)$ below $(Q, f)$ is below $(P, e)$ by transitivity. Its uniqueness forces $c = b$ .

F1F2step 1.1∎

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

Maximal Brauer pairs exist and are conjugate Theorem

Dependency tree · two levels

10 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, Lemma 2.32 and Theorem 2.33, pp.18–19; general-field lifting proved locally (standard reference, not scraped)