Conjugacy Action on Group Elements is Group Action

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

$G$ acts on itself by the rule:
 * $\forall g, h \in G: g * h = g \circ h \circ g^{-1}$

Also: where $C_x$ is the conjugacy class of $x$.
 * $\operatorname{Orb} \left({x}\right) = C_x$

Proof
Clearly GA-1 is fulfilled as $e * x = x$.

GA-2 is shown to be fulfilled thus:

$\operatorname{Orb} \left({x}\right) = C_{x}$ follows from the definition of the conjugacy class.