DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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 external semidirect product $N ⋊_{α} H$

Definition

Let $N$ and $H$ be groups (Group and abelian group), and let $α : H \to Aut (N)$ be an action by automorphisms (An action of a group $H$ on a group $N$ by automorphisms). The external semidirect product $N ⋊_{α} H$ is the set $N \times H$ with multiplication

$(n, h) (n^{'}, h^{'}) = (n α_{h} (n^{'}), h h^{'}) .$

When the action is clear, the subscript is omitted.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 18 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Keith Conrad, Semidirect Products (standard reference, not scraped)

The external semidirect product N⋊αHN⋊α​H