External wreath product with diagonal action

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This article describes a product notion for groups. See other related product notions for groups.

Definition

This definition uses the left action convention.

Let G be any abstract group and H be a group along with a homomorphism \rho:H \to \operatorname{Sym}(S) for some set S (in other words, we are given a permutation representation, or group action, of H). Let K be a group along with a homomorphism \alpha:K \to \operatorname{Aut}(G).

Then, the external wreath product with diagonal action of G with H as the wreathing group (relative to \rho) and K as the diagonally acting group is defined as the external semidirect product of the group G^S by H \times K, where a pair (h,k) \in H \times K sends f:S \to G to the function \alpha(k) \circ f \circ \rho(h^{-1}).

The group G is termed the base of the wreath product with diagonal action.

The wreath product with diagonal action can be viewed as follows: consider the group G^S obtained as a direct product of |S| copies of G labeled by the elements of S. The wreath product with diagonal action is the semidirect product of G^S by the action of H \times K, where K acts by automorphisms on each coordinate via \alpha, and H permutes the coordinates via the action \rho on S.

Note that in the case that K is the trivial group, we simply get the external wreath product of G by H.

Related notions