# Action of wreath product on Cartesian product

From Groupprops

## Definition

Suppose both and are being viewed along with specific group actions of each, i.e., they are both being viewed as groups of permutations, with acting on a set and acting on a set . Then, comes equipped with a natural action on the Cartesian product as follows.

First, recall that is the external semidirect product . To specify an action of this, we will specify how and act; then, via the equivalence of internal and external semidirect product, we would have defined the action of a generating set and hence of all of . Here is the action:

- acts on as follows: An element of is a function . For an element , the action of on is obtained as where is the action of the element on . In other words, it permutes each of the fibers according to the -value on that fiber.
- acts on as follows: An element sends to where denotes the image of under the action of .

Note that we need to verify a compatibility condition to show that this is well defined.