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 $h$.

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