The orbit-stabilizer theorem is a combinatorial result in group theory.
Let be a group acting on a set . For any , let denote the stabilizer of , and let denote the orbit of . The orbit-stabilizer theorem states that
Proof. Without loss of generality, let operate on from the left. We note that if are elements of such that , then . Hence for any , the set of elements of for which constitute a unique left coset modulo . Thus The result then follows from Lagrange's Theorem.