Group Acts on Itself

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

Then $\left({G, \circ}\right)$ acts on itself by the rule:

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

The orbit and stabilizer of an element $x \in G$ are as follows:
 * $\operatorname{Orb} \left({x}\right) = G$.
 * $\operatorname{Stab} \left({x}\right) = \left\{{e}\right\}$.

Proof
Follows directly from the group axioms and the definition of a group action.


 * $\operatorname{Orb} \left({x}\right) = G$ by the Latin Square Property.


 * $\operatorname{Stab} \left({x}\right) = \left\{{e}\right\}$ follows from the fact that $g \circ x = x \implies g = e$.