---
type: "concept"
title: "Burnside's lemma"
slug: "burnsides-lemma"
language: "en"
translationGroupId: "cmrtoobhk000flf01806uebqk"
domain: "Algebra"
status: "usable"
aliases: []
lastEditedBy: "ancient-tree"
---

Let $G$ be a [[Finite group|finite group]] [[Group action|acting]] on a set $X$. Then the number of [[Orbit of a group action|orbits]] is given by :
$$|X / G|=\frac{1}{|G|} \sum_{g \in G}|\operatorname{Fix}(g)|$$
where $\operatorname{Fix}(g)$ is the set of elements of $X$ fixed by $g$.