Action of wreath product on Cartesian product
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.