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.

An internal semidirect product and a complement to a normal subgroup

Definition

Let $N$ and $H$ be subgroups (Subgroup) of a group $G$ . The group $G$ is the internal semidirect product of $N$ by $H$ when

$N ⊴ G, G = N H, N \cap H = {1} .$

Here $N ⊴ G$ means that $N$ is normal in $G$ (Normal subgroup: invariance under conjugation). In this situation $H$ is called a complement to $N$ in $G$ .

Depends on

Used by

Recognition theorem: G=NH with N is normal in G, N∩ H=1 exactly realises an external semidirect product Theorem

Dependency tree · next 3 levels

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