Stabilizer of Coset Action on Set of Subgroups

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 $H \in \mathcal P \left({G}\right)$ be a subgroup of $G$.

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

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

Then the stabilizer of $H$ in $\mathcal P \left({G}\right)$ is $H$ itself:
 * $\operatorname{Stab} \left({H}\right) = H$

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


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

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

Also see

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