# Action of wreath product on function space

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.