Burnside's Lemma

The number of orbits equals the average number of fixed points

Burnside’s Lemma

Burnside’s Lemma (Cauchy–Frobenius). Let $G$ be a finite group acting on a finite set $X$ via a group action . For $g \in G$ , let

\operatorname{Fix}(g)=\{x\in X : g\cdot x = x\}

be the fixed-point set of $g$ . Then the number of orbits of the action is

|X/G| = \frac{1}{|G|}\sum_{g\in G} |\operatorname{Fix}(g)|.

Burnside’s lemma is a standard counting tool: instead of counting orbits directly, it averages easily computed fixed-point counts. It is frequently used in enumeration problems involving symmetries.