Restricted external wreath product
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.
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.