Subset Product Action is Group Action

Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $\powerset G$ be the power set of $\struct {G, \circ}$.

For any $S \in \powerset G$ and for any $g \in G$, the subset product action:
 * $\forall g \in G: \forall S \in \powerset G: g * S = g \circ S$

is a group action.

Proof
Let $g \in G$.

First we note that since $G$ is closed, and $g \circ S$ consists of products of elements of $G$, it follows that:
 * $g * S \subseteq G$

Next we note:
 * $e * S = e \circ S = \set {e \circ s: s \in S} = \set {s: s \in S} = S$

and so is satisfied.

Now let $g, h \in G$.

We have:

and so is satisfied.

Hence the result.

Also see

 * Stabilizer of Subset Product Action on Power Set
 * Stabilizer of Coset Action on Set of Subgroups
 * Orbit of Subgroup under Coset Action is Coset Space