DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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.

The graph subgroup attached to a map into a semidirect product

Definition

Let $G$ act on a group $M$ , and let $M ⋊ G$ be the corresponding external semidirect product ( The external semidirect product $N ⋊_{α} H$ ).

For any function $z : G \to M$ , its graph subgroup candidate is the subset

$Γ_{z} : = {(z (g), g) : g \in G} \subseteq M ⋊ G .$

It is called the graph subgroup of $z$ when this subset is actually a subgroup.

Depends on

The external semidirect product $N\rtimes_\alpha H$

Used by

Dependency tree · two levels

3 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

David A. Craven, Finite Group Theory (standard reference, not scraped)