Orbit of Conjugacy Action on Subgroup is Set of Conjugate Subgroups

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 $H$ defined as:


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

Then the orbit $\operatorname{Orb} \left({H}\right)$ of $H$ in $\mathcal P \left({G}\right)$ is $H$ is the set of subgroups conjugate to $H$.

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

from the definition.

The result follows by definition of conjugate subgroup.

Also see

 * Conjugacy Action on Subgroups is Group Action
 * Stabilizer of Conjugacy Action on Subgroup is Normalizer