# External wreath product with diagonal action

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 $H$.