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 $$g \circ x = x \implies g = e$$.