# Restricted external wreath product

## Definition

### For an abstract group and a group of permutations

This definition uses the left action convention.

Let be any abstract group and be a group along with a homomorphism for some set (in other words, we are given a permutation representation, or group action, of ). Then, the **restricted external wreath product** of by (relative to ) is defined as the external semidirect product of the restricted external direct product (which can be thought of as functions of finite support from to ) by , where an element sends to the function .

This wreath product is typically denoted as:

The group is termed the **base** of the wreath product.

The wreath product can also be viewed as follows: The group is the restricted external direct product of copies of , and the action of is to permute the coordinates by the permutation . 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 is a finite set, these two notions are equivalent.

### For two abstract groups

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