Conjugacy Action on Group Elements is Group Action/Proof 2

Proof
By Subset Product Action is Group Action and Group is Subgroup of Itself, it follows that for any $G \le G$ and for any $g \in G$, the conjugacy action:
 * $g * G := g \circ G \circ g^{-1}$

is a group action.

Hence it follows that
 * $\forall g, h \in G: g * h = g \circ h \circ g^{-1}$

is a group action, as required.