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

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Block bimodule has a diagonal vertex

Statement

For every block $b$ of $k G$ , a vertex of its indecomposable double module $k G b$ is conjugate in $G \times G$ to $Δ D$ for some $p$ -subgroup $D \leq G$ .

Facts & Assumptions

Given: A block bimodule over the residue field of a splitting $p$ -modular system.

[F1]

The whole bimodule $k G$ is induced from $Δ G$ . (Group algebra bimodule is induced from the diagonal)

[F2]

A vertex of an indecomposable relatively $H$ -projective module is conjugate into $H$ . (Relative projectivity mackey intersections for finite modules)

[F3]

An indecomposable finite-dimensional module has a vertex. (Vertices exist for indecomposable modules, are conjugate in G, and sources are conjugate by the appropriate normalizer)

Proof

technique · direct

1.1

The nonzero indecomposable block bimodule is a direct summand of $k G$ , so [F1] makes it relatively $Δ G$ -projective. Choose a vertex $Q$ by [F3]; [F2] conjugates it into $Δ G$ .

F1F2F3

2.1

Projection of this conjugate vertex onto the first factor is a subgroup $D \leq G$ ; every element of $Δ G$ has the form $(d, d)$ , so the conjugate vertex equals $Δ D$ . Projection is an isomorphism there, hence $D$ is a $p$ -subgroup. The subgroup obtained is a vertex, since conjugation preserves relative projectivity and inclusion-minimality.

step 1.1∎

Sources

Webb, A Course in Finite Group Representation Theory, §§5.2, 11.3, 11.6, 12.3–12.5; especially Lemma 12.4.4 and Theorem 12.4.5, pp.240–241. Local argument and conventions as displayed above.

Depends on

Used by

Defect group and numerical defect of a block Definition
Centralizer containment makes block induction well-defined Lemma
Induced blocks have controlled defect Lemma

Dependency tree · two levels

8 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, §§5.2, 11.3, 11.6, 12.3–12.5; especially Lemma 12.4.4 and Theorem 12.4.5, pp.240–241 (standard reference, not scraped)