# Stabilizer of Conjugacy Action on Subgroup is Normalizer

## Theorem

Let $\struct {G, \circ}$ 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 $\powerset G$ is given by:

$\Stab H = \map {N_G} H$

where $\map {N_G} H$ is the normalizer of $H$ in $G$.

## Proof

We have that:

$\Stab H = \set {g \in G: g \circ H \circ g^{-1} = H}$

which is precisely how the normalizer is defined.

$\blacksquare$