Stabilizer of Element under Conjugacy Action is Centralizer

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

Let $*$ be the conjugacy 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 conjugacy action is:
 * $\Stab x = \map {C_G} x$

where $\map {C_G} x$ is the centralizer of $x$ in $G$.

Proof
From the definition of centralizer:
 * $\map {C_G} x = \set {g \in G: g \circ x = x \circ g}$

Then:

Also see

 * Conjugacy Action on Group Elements is Group Action
 * Orbit of Element under Conjugacy Action is Conjugacy Class