# Stabilizer of Coset Action on Set of Subgroups

## Theorem

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

Let $\powerset G$ denote the power set of $G$.

Let $\HH \subseteq \powerset G$ denote the set of subgroups of $G$.

Let $*$ be the subset product action on $\HH \subseteq \powerset G$ defined as:

$\forall g \in G: \forall H \in \HH: g * H = g \circ H$

where $g \circ H$ is the (left) coset of $g$ by $H$.

Then the stabilizer of $H$ in $\powerset G$ is $H$ itself:

$\Stab H = H$

## Proof

From the definition of Stabilizer of Subset Product Action on Power Set:

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

The result follows from Left Coset Equals Subgroup iff Element in Subgroup.

$\blacksquare$