Orbit-stabilizer theorem

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 right. We note that if are elements of such that , then $\alpha^{-1} \beta \in \stab(i)$ (Error compiling LaTeX. Unknown error_msg). 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.

See also

• Burnside's Lemma
• Orbit
• Stabilizer