Stabilizer of Subset Product Action on Power Set

Theorem
Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Let $\mathcal P \left({G}\right)$ be the power set of $\left({G, \circ}\right)$.

Let $S \in \mathcal P \left({G}\right)$.

Let $*$ be the group action on $S$ defined as:
 * $\forall g \in G: g * S = g \circ S$

where $g \circ S$ is the subset product $\left\{{g}\right\} \circ S$.

Then the stabilizer of $S$ in $\mathcal P \left({G}\right)$ is the set:
 * $\operatorname{Stab} \left({S}\right) = \left\{{g \in G: g \circ S = S}\right\}$

Proof
From the definition of stabilizer:


 * $\operatorname{Stab} \left({S}\right) = \left\{{g \in G: g * S = S}\right\}$

The result follows from the definition of the group action $*$ given.

Also see

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