Restricted external wreath product

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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.