Orbit of Element of Group Acting on Itself is Group

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

Let $*$ be the group action of $\left({G, \circ}\right)$ on itself by the rule:


 * $\forall g, h \in G: g * h = g \circ h$

Then the orbit of an element $x \in G$ is given by:
 * $\operatorname{Orb} \left({x}\right) = G$

Proof
Let $y \in G$.

Then:

Hence the result.

Also see

 * Group Acts on Itself
 * Stabilizer of Element of Group Acting on Itself is Trivial