# Statements > [!NOTE] Theorem > Let $G$ be a [[Groups|group]] [[Group action|acting]] on a set $X.$ Let $X/G$ denote the set of [[Orbit under Group Action|orbits]]. Then there exists a bijection $G\times X/G\leftrightarrow \bigsqcup_{g\in G}X^g$where for $g\in G,$ $X^{g}=\{ x\in X: g\cdot x= x \}$ # Proofs ###### Proof Follows from [[Orbit-Stabilizer Theorem]]. # Applications # Bibliography 1. Bogart, K. (1991). An obvious proof of Burnside’s lemma. Amer. Math. Monthly. 98(10): 927–928.