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 $C_G \left({x}\right)$ is the centralizer of $x$ in $G$.
 * $\operatorname{Stab} \left({x}\right) = C_G \left({x}\right)$

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:

$\operatorname{Stab} \left({x}\right) = C_G \left({x}\right)$ follows from the definition of centralizer: $C_G \left({x}\right) = \left\{{g \in G: g \circ x = x \circ g}\right\}$.

Furthermore, since the powers of $x$ commute with $x$, $\left \langle {x} \right \rangle \in C_G \left({x}\right)$.

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