Conjugacy Action is not Transitive

Theorem
Let $\left({G, \circ}\right)$ be a non-trivial 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 not a transitive group action.

Proof
Proof by Counterexample:

For $G$ to be a transitive group action, the orbit of any element of $G$ needs to be the whole of $G$.

Take $h = e$.

Then:

Thus by definition of orbit:
 * $\operatorname{Orb} \left({e}\right) = \left\{{e}\right\}$

Only when $G$ is the trivial group, that is: $G = \left\{{e}\right\}$, does $\operatorname{Orb} \left({e}\right) = G$.

Hence the result by definition of transitive group action.