DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

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The imprimitive wreath product of permutation groups

Definition

Let $H$ act on a set $B$ , and let $K$ act on a set $Σ$ , both by left actions (Left group actions, transitive actions, and faithful actions).

Write $H^{Σ} : = {f : Σ \to H}$ with pointwise multiplication. The action of $K$ on $H^{Σ}$ by automorphisms is $(k \cdot f) (σ) : = f (k^{- 1} \cdot σ) .$ Using An action of a group $H$ on a group $N$ by automorphisms and The external semidirect product $N ⋊_{α} H$ , form the semidirect product $H^{Σ} ⋊ K .$

The imprimitive wreath product of the two permutation groups is this semidirect product, written $H ≀_{Σ} K : = H^{Σ} ⋊ K .$

It acts on $B \times Σ$ by

(f, k) \cdot (b, σ) : = (f (k \cdot σ) \cdot b, k \cdot σ) .

Indeed, if

(f, k) (f^{'}, k^{'}) = (f \cdot (k \cdot f^{'}), k k^{'}),

then the first coordinate at $(b, σ)$ becomes $(f \cdot (k \cdot f^{'})) (k k^{'} \cdot σ) \cdot b = f (k k^{'} \cdot σ) \cdot (f^{'} (k^{'} \cdot σ) \cdot b),$ which is exactly what one gets by first applying $(f^{'}, k^{'})$ and then $(f, k)$ .

Depends on

Used by

Dependency tree · two levels

8 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

J. S. Milne, Group Theory, Chapter 4 (standard reference, not scraped)
K. Conrad, Semidirect Products (standard reference, not scraped)