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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Normal p core lies in every block defect group

Statement

For a finite group $G$ , its largest normal $p$ -subgroup $O_{p} (G)$ lies in every block defect group over a field of characteristic $p$ .

Facts & Assumptions

Given: A finite group $G$ and a defect group $D$ .

[F1]

A defect group is an intersection of two Sylow $p$ -subgroups. (Block defect is an intersection of two sylow subgroups)

[F2]

A $p$ -subgroup can be 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

1.1given

For finite $K$ , the product of two normal $p$ -subgroups $N_{1}, N_{2}$ is normal and is a subgroup; its order is $∣ N_{1} ∣ ∣ N_{2} ∣ / ∣ N_{1} \cap N_{2} ∣$ , a power of $p$ . The order formula follows because each element of the product has exactly $∣ N_{1} \cap N_{2} ∣$ representations as $n_{1} n_{2}$ . There are finitely many subgroups of $K$ , so their finite product defines the largest normal $p$ -subgroup $O_{p} (K)$ , including the trivial subgroup. If $T$ is Sylow and $N ⊴ K$ is a $p$ -subgroup, the same order formula makes $N T$ a $p$ -subgroup containing $T$ ; maximality of $T$ forces $N T = T$ . Thus $N \leq T$ .

2.1F1F2step 1.1∎

Choose a Sylow $P$ containing $D$ by [F2]. By [F1], $D = P \cap h P h^{- 1}$ for some $h$ . Apply step 1.1 to both Sylow subgroups to obtain $O_{p} (G) \leq P \cap h P h^{- 1} = D$ .

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

Nagao decomposition for restriction to a centralizer Theorem

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)