Restricted external wreath product

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 defining ingredient::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.