Stabilizer of Element under Conjugacy Action is Centralizer

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

Let $*$ be the group action on $G$ defined by the rule:
 * $\forall g, h \in G: g * h = g \circ h \circ g^{-1}$

Let $x \in G$.

Then the stabilizer of $x$ under this group action is:
 * $\operatorname{Stab} \left({x}\right) = C_G \left({x}\right)$

where $C_G \left({x}\right)$ is the centralizer of $x$ in $G$.

Proof
From the definition of centralizer:
 * $C_G \left({x}\right) = \left\{{g \in G: g \circ x = x \circ g}\right\}$

Then:

Furthermore, since the powers of $x$ commute with $x$, it follows that:
 * $\left \langle {x} \right \rangle \in C_G \left({x}\right)$