Conjugacy Action on Subgroups is Group Action

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

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

For any $H \le G$ and for any $g \in G$, the conjugacy action:
 * $g * H := g \circ H \circ g^{-1}$

is a group action.

Proof
Clearly is fulfilled as $e * H = H$.

is shown to be fulfilled thus:

Also see

 * Stabilizer of Conjugacy Action on Subgroup is Normalizer
 * Orbit of Conjugacy Action on Subgroup is Set of Conjugate Subgroups