Orbit-Stabilizer Theorem

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

Let $\operatorname{Orb} \left({x}\right)$ be the orbit of $x$.

Let $\operatorname{Stab} \left({x}\right)$ be the stabilizer of $x$ by $G$.

Let $\left[{G : \operatorname{Stab} \left({x}\right)}\right]$ be the index of $\operatorname{Stab} \left({x}\right)$ in $G$.

Then:
 * $\displaystyle \left|{\operatorname{Orb} \left({x}\right)}\right| = \left[{G : \operatorname{Stab} \left({x}\right)}\right] = \frac {\left|{G}\right|} {\left|{\operatorname{Stab} \left({x}\right)}\right|}$