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

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Vertices of modules in a block lie in a defect group

Statement

If $M$ is a nonzero indecomposable finite-dimensional $k G$ -module with $b M = M$ and $D$ is a defect group of $b$ , every vertex of $M$ is conjugate into $D$ .

Facts & Assumptions

Given: $M, b, D$ as in the statement.

[F1]

A defect group supplies $b = {Tr}_{D}^{G} (a)$ for $a \in (k G b)^{D}$ . (Block relative trace characterizes diagonal projectivity)

[F2]

A trace equal to the identity proves relative projectivity. (Higman's criterion characterizes relative projectivity through the relative trace idempotent test)

[F3]

Vertices of a relatively $D$ -projective indecomposable module are conjugate into $D$ . (Relative projectivity mackey intersections for finite modules)

Proof

technique · direct

1.1

Choose $a$ from [F1]. Its multiplication action on $M$ is $D$ -linear because $a$ commutes with $D$ . The trace of that endomorphism is multiplication by ${Tr}_{D}^{G} (a) = b$ , hence is ${i d}_{M}$ since $b M = M$ .

2.1

Higman makes $M$ relatively $D$ -projective. Vertex containment in [F3] then places each vertex in a conjugate of $D$ , or equivalently conjugates it into $D$ .

F2F3step 1.1∎

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

Principal block has sylow defect Proposition
Brauer–Green block compatibility Theorem
Defect zero blocks are simple algebras 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

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)