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

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The holomorph $Hol (G) = G ⋊ Aut (G)$

Definition

The holomorph of a group $G$ is the external semidirect product ( The external semidirect product $N ⋊_{α} H$ )

$Hol (G) = G ⋊ Aut (G),$

where $Aut (G)$ is the group supplied by The automorphisms of a group form a group under composition and acts on $G$ by evaluation. Thus

$(g, α) (h, β) = (g α (h), α β) .$

Depends on

Used by

Hol(C₂× C₂)≅ S₄ Example
Hol(C₈) is the group of affine maps x↦ ax+b with a∈{1,3,5,7} Example
The holomorph acts faithfully on G by affine permutations x↦ gα(x) Proposition

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 18 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

Peter J. Cameron, The Holomorph of a Group (standard reference, not scraped)

The holomorph Hol⁡(G)=G⋊Aut⁡(G)Hol(G)=G⋊Aut(G)