Action of wreath product on Cartesian product

Definition

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

First, recall that $G \wr S$ is the external semidirect product $G^S \rtimes H$ . To specify an action of this, we will specify how $G^S$ and $H$ 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 $G \wr H$ . Here is the action:

$G^S$ acts on $A \times S$ as follows: An element of $G^S$ is a function $f:S \to G$ . For an element $(a,s)$ , the action of $f$ on $(a,s)$ is obtained as $(f(s) \cdot a, s)$ where $f(s) \cdot a$ is the action of the element $f(s) \in G$ on $A$ . In other words, it permutes each of the fibers according to the $f$ -value on that fiber.
$H$ acts on $A \times S$ as follows: An element $h \in H$ sends $(a,s) \in A \times S$ to $(a,h \cdot s)$ where $h \cdot s$ denotes the image of $s$ under the action of $h$ .