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 $H$ .

Related notions