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

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Block defect groups are p radical

Statement

Every block defect group satisfies $D = O_{p} (N_{G} (D))$ .

Facts & Assumptions

Given: A defect group $D$ of a block of a finite group $G$ over a field of characteristic $p$ .

[F1]

For every Sylow $P \geq D$ there is $h \in C_{G} (D)$ with $D = P \cap h P h^{- 1}$ . (Block defect is an intersection of two sylow subgroups)

[F2]

Every $p$ -subgroup is contained in a Sylow subgroup. (Sylow II: in a finite group every $p$ -subgroup lies in a conjugate of any Sylow $p$ -subgroup, and the Sylow $p$ -subgroups form a single conjugacy class)

Proof

technique · direct

1.1

Put $N = N_{G} (D)$ . Choose a Sylow subgroup $T$ of $N$ containing $D$ , and a Sylow subgroup $P$ of $G$ containing $T$ , by [F2]. Then $P \cap N$ is a $p$ -subgroup of $N$ containing its Sylow subgroup $T$ , hence equals $T$ . Take $h \in C_{G} (D) \subseteq N$ from [F1]. Intersecting $D = P \cap h P h^{- 1}$ with $N$ gives $D = (P \cap N) \cap h (P \cap N) h^{- 1} = T \cap h T h^{- 1}$ .

F1F2

2.1

The product of normal $p$ -subgroups of a finite group is a normal $p$ -subgroup: for two factors its order is the product of their orders divided by the intersection order, by counting representations of a product element. A finite product over all such subgroups therefore defines $O_{p} (N)$ . If $V$ is Sylow in $N$ , normality makes $O_{p} (N) V$ a $p$ -subgroup containing $V$ , hence equal to $V$ . Thus $O_{p} (N)$ lies in both $T$ and $h T h^{- 1}$ , so in $D$ by step 1.1. Conversely $D ⊴ N$ by the definition of its normalizer and $D$ is a $p$ -group, so $D \leq O_{p} (N)$ . Both inclusions prove equality.

step 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

Every block contains a module whose vertex is a full defect group Corollary

Dependency tree · two levels

13 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)