DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)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.

An action of a group $H$ on a group $N$ by automorphisms

Definition

An action of a group $H$ on a group $N$ by automorphisms is a homomorphism

$α : H ⟶ Aut (N) .$

Here automorphisms are those of Group isomorphisms, automorphisms and the set $Aut (G)$ . Writing $α_{h} = α (h)$ , this means that every $α_{h}$ is an automorphism of $N$ , $α_{h k} = α_{h} \circ α_{k}$ , and $α_{1} = {id}_{N}$ . Equivalently, by Actions of $G$ on $X$ correspond exactly to homomorphisms $G \to Sym (X)$ , it is a group action (Left group actions, transitive actions, and faithful actions) on the underlying set of $N$ for which every acting permutation is an automorphism.

Depends on

Used by

The external semidirect product N rtimes_α H Definition
The semidirect-product multiplication makes N× H a group Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 27 results over 11 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)

An action of a group HH on a group NN by automorphisms