DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-27

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.

Minimal normal subgroups and the socle of a finite group

Definition

Let $G$ be a finite group.

A nontrivial normal subgroup $N ⊴ G$ is a minimal normal subgroup of $G$ if the only normal subgroups of $G$ contained in $N$ are $1$ and $N$ itself.

The socle of $G$ is the subgroup

$soc (G) : = ⟨ N : N ⊴ G is a minimal normal subgroup ⟩,$

that is, the subgroup generated in the sense of The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups by all minimal normal subgroups of $G$ .

Depends on

Used by

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

James E. Humphreys, A Course in Group Theory, Chapter 16 (standard reference, not scraped)