Orbit_Stabilizer_Theorem_v2

Let $G$ be a group which acts on a finite set $X$. Let $x \in X$. 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 $\order {\Orb x} = \index G {\Stab x} = \dfrac {\order G} {\order {\Stab x} }$