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

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Defect groups of a block are conjugate

Statement

All defect groups of a block $b$ are conjugate in $G$ . Every conjugate of a defect group is again a defect group, so the numerical defect is well-defined.

Facts & Assumptions

Given: Defect groups $D, E$ for the same block.

[F1]

Defect groups are the components of diagonal vertices. (Defect group and numerical defect of a block)

[F2]

Vertices of one indecomposable module are conjugate in its ambient group. (Vertices exist for indecomposable modules, are conjugate in G, and sources are conjugate by the appropriate normalizer)

Proof

technique · direct

1.1

By [F2], some $(x, y) \in G \times G$ satisfies $(x, y) Δ D (x, y)^{- 1} = Δ E$ . Projecting to the first coordinate gives $x D x^{- 1} = E$ . Thus the groups have equal orders, proving independence of the numerical defect.

F1F2

2.1

Conversely, conjugation by $(x, x)$ takes $Δ D$ to $Δ (x D x^{- 1})$ and preserves its vertex property. Hence $x D x^{- 1}$ is a defect group by [F1].

F1F2∎

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 groups are maximal Brauer support Theorem

Dependency tree · two levels

6 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)