Restricted external wreath product

Definition

For an abstract group and a group of permutations

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 $H$). Then, the restricted external wreath product of $G$ by $H$ (relative to $\rho$) is defined as the external semidirect product of the restricted external direct product $G^S$ (which can be thought of as functions of finite support from $S$ to $G$) by $H$, where an element $h \in H$ sends $f:S \to G$ to the function $f \circ \rho(h^{-1})$.

This wreath product is typically denoted as:

$G \wr H$

The group $G$ is termed the base of the wreath product.

The wreath product can also be viewed as follows: The group $G^S$ is the restricted external direct product of $|S|$ copies of $G$, and the action of $h \in H$ is to permute the $|S|$ coordinates by the permutation $\rho(h)$. The significance of restricted direct product is that only finitely many coordinates can be non-identity elements.

There is a related notion of (unrestricted) external wreath product, where we use the unrestricted external direct product. Note that when $S$ is a finite set, these two notions are equivalent.

For two abstract groups

Let $G,H$ be abstract groups. Then, the restricted external wreath product of $G$ by $H$ is typically understood as the wreath product where the action of $H$ is taken to be the regular group action on itself as a set. In other words, the homomorphism $\rho$ is the natural embedding arising via Cayley's theorem. This wreath product is also termed the restricted regular wreath product.