Orbit-Stabilizer Theorem

Theorem
Let $G$ be a group which acts on a finite set $X$.

Let $\Orb x$ denote the orbit of $x$.

Let $\Stab x$ denote the stabilizer of $x$ by $G$.

Let $\index G {\Stab x}$ denote the index of $\Stab x$ in $G$.

Then:
 * $\order {\Orb x} = \index G {\Stab x} = \dfrac {\order G} {\order {\Stab x} }$