DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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

Frobenius semidirect product decomposition Corollary
The centre of an extraspecial p-group has no complement Corollary
A solvable group that is not an M-group Counterexample
Normal p complement and p nilpotent group Definition
Affine linear Frobenius groups over finite fields Example
S₃ as a Frobenius group Example
FALSE: the centre of an extraspecial p-group has a complement False statement
Equivalent forms of having a normal p complement Proposition
Frobenius groups and fixed point free actions Proposition
Burnside normal p complement theorem Theorem
Recognition theorem: G=NH with N is normal in G, N∩ H=1 exactly realises an external semidirect product Theorem
The little group method for a semidirect product with abelian kernel Theorem

Dependency tree · two levels

7 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

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