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 $S \in \powerset G$, we define:
 * $\forall g \in G: g * S = g \circ S$

where $g \circ S$ is the subset product $\set g \circ S$.

This is a group action.

Proof
The fact that this is a group action follows directly from the definitions:

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 Power Set
 * Orbit of Subgroup under Coset Action is Coset Space