Orbit of Element under Conjugacy Action is Conjugacy Class

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 orbit of $x$ under this group action is:
 * $\operatorname{Orb} \left({x}\right) = C_x$

where $C_x$ is the conjugacy class of $x$.

Proof
Follows from the definition of the conjugacy class.

Also see

 * Stabilizer of Element under Conjugacy Action is Centralizer