PropositionStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The holomorph acts faithfully on $G$ by affine permutations $x \mapsto g α (x)$

Statement

For every group $G$ , the rule

$ρ (g, α) (x) = g α (x)$

defines a faithful action of $Hol (G)$ on the underlying set of $G$ . Equivalently, $ρ$ embeds $Hol (G)$ in $Sym (G)$ .

Facts & Assumptions

Given: A group $G$ and its holomorph.

[L1]

The holomorph multiplication is $(g, α) (h, β) = (g α (h), α β)$ ( The holomorph $Hol (G) = G ⋊ Aut (G)$ ).

[L2]

A group action is a rule satisfying the identity and compatibility laws, and it is faithful when only the identity acts trivially (Left group actions, transitive actions, and faithful actions).

[L3]

Group actions on a set correspond to homomorphisms into its symmetric group (Actions of $G$ on $X$ correspond exactly to homomorphisms $G \to Sym (X)$ , The symmetric group $Sym (X)$ : the bijections of a set $X$ under composition).

Proof

technique · direct

1.1

Each map $x \mapsto g α (x)$ is a permutation, with inverse $x \mapsto α^{- 1} (g^{- 1} x)$ .

L1algebra

1.2

Composition gives $ρ (g, α) (ρ (h, β) (x)) = g α (h) (α β) (x)$ , which equals $ρ ((g, α) (h, β)) (x)$ by [L1]. Therefore $ρ$ is an action, equivalently a homomorphism to $Sym (G)$ , by [L2] and [L3].

L1L2L3

2.1

If $ρ (g, α)$ is the identity permutation, evaluation at $1_{G}$ gives $g = 1_{G}$ . Then $α (x) = x$ for every $x \in G$ , so $α = {id}_{G}$ . Thus the action is faithful by [L2], and the corresponding homomorphism is injective: equality of two images reduces, after multiplying by an inverse, to this identity case.

step 1.2L2L3algebra∎

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 affine group of the real line is ℝrtimesℝ^× Example
A permutation group with a regular normal subgroup G embeds in Hol(G) Proposition

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 33 results over 15 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 acts faithfully on GG by affine permutations x↦gα(x)x↦gα(x)