# Stabilizer of Subset Product Action on Power Set

## Theorem

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

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

Let $*: G \times \powerset G \to \powerset G$ be the subset product action on $\powerset G$ defined as:

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

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

Then the stabilizer of $S$ in $\powerset G$ is the set:

$\Stab S = S$

## Proof

From the definition of stabilizer:

$\Stab S = \set {g \in G: g * S = S}$

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

$\blacksquare$