Conjugacy Action on Abelian Group is Trivial

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

Let $*: G \times G \to G$ be the conjugacy group action:
 * $\forall g, h \in G: g * h = g \circ h \circ g^{-1}$

Then $*$ is a trivial group action.

Proof
For $G$ to be a trivial group action, the orbit of any element of $G$ is a singleton containing only that element.

Take $h \in G$.

Then:

Thus by definition of orbit:
 * $\Orb h = \set h$

Hence the result by definition of trivial group action.