Stabilizer of Conjugacy Action on Subgroup is Normalizer

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

Let $X$ be the set of all subgroups of $G$.

Let $*$ be the conjugacy action on $H$:


 * $\forall g \in G, H \in X: g * H = g \circ H \circ g^{-1}$

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

where $N_G \left({H}\right)$ is the normalizer of $H$ in $G$.

Proof
We have that:
 * $\operatorname{Stab} \left({H}\right) = \left\{{g \in G: g \circ H \circ g^{-1} = H}\right\}$

which is precisely how the normalizer is defined.

Also see

 * Conjugacy Action on Subgroups is Group Action
 * Orbit of Conjugacy Action on Subgroup is Set of Conjugate Subgroups