Action of wreath product on function space

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 set $A^{S}$ of all functions from $S$ to $A$.

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 $A^{S}$ as follows: Given a function $f:S\to G$ and a function $\theta :S\to A$, define $f\cdot \theta$ as the function $S\to A$ given by $s\mapsto f(s)\cdot \theta (s)$, where we use the action of $G$ on $A$. One way of thinking of this is that the action is coordinate-wise or point-wise.
• $H$ acts on $A^{S}$ as follows: Given an element $h\in H$ and a function $\theta :S\to A$, we define $h\cdot \theta$ as the map $s\mapsto \theta (h^{-1}\cdot s)$ where we use the action of $H$ on $S$ on the inside.