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Burnside's Lemma

The number of orbits equals the average number of fixed points

Burnside's Lemma (Cauchy–Frobenius). Let GG be a finite acting on a finite set XX via a . For g∈Gg \in G, let

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

be the of gg. Then the number of of the action is

∣X/G∣=1∣G∣∑g∈G∣Fix⁡(g)∣.|X/G| = \frac{1}{|G|}\sum_{g\in G} |\operatorname{Fix}(g)|.
Remarks

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.