Burnside's lemma
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Revision 542
7/21/2026, 1:06:28 PM · Ancient Tree
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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 :2
$$|X / G|=\frac{1}{|G|} \sum_{g \in G}|\operatorname{Fix}(g)|$$3
where $\operatorname{Fix}(g)$ is the set of elements of $X$ fixed by $g$.Revision 529
7/20/2026, 8:36:10 PM · Ancient Tree
Concept edited
Compare with revision 528No text changes
1
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 :2
$$|X / G|=\frac{1}{|G|} \sum_{g \in G}|\operatorname{Fix}(g)|$$3
where $\operatorname{Fix}(g)$ is the set of elements of $X$ fixed by $g$.Revision 528
7/20/2026, 8:34:57 PM · Ancient Tree
Concept created
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 $g$.