# External wreath product with diagonal action

This article describes a product notion for groups. See other related product notions for groups.

## Definition

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 ). Let be a group along with a homomorphism .

Then, the **external wreath product with diagonal action** of with as the wreathing group (relative to ) and as the diagonally acting group is defined as the external semidirect product of the group by , where a pair sends to the function .

The group is termed the **base** of the wreath product with diagonal action.

The wreath product with diagonal action can be viewed as follows: consider the group obtained as a direct product of copies of labeled by the elements of . The wreath product with diagonal action is the semidirect product of by the action of , where acts by automorphisms on each coordinate via , and permutes the coordinates via the action on .

Note that in the case that is the trivial group, we simply get the external wreath product of by .