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- Burnside's_lemma abstract "Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy-Frobenius lemma or the orbit-counting theorem, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. Its various eponyms include William Burnside, George Pólya, Augustin Louis Cauchy, and Ferdinand Georg Frobenius. The result is not due to Burnside himself, who merely quotes it in his book 'On the Theory of Groups of Finite Order', attributing it instead to Frobenius (1887).In the following, let G be a finite group that acts on a set X. For each g in G let Xg denote the set of elements in X that are fixed by g. Burnside's lemma asserts the following formula for the number of orbits, denoted |X/G|:Thus the number of orbits (a natural number or +∞) is equal to the average number of points fixed by an element of G (which is also a natural number or infinity). If G is infinite, the division by |G| may not be well-defined; in this case the following statement in cardinal arithmetic holds:".
- Burnside's_lemma thumbnail Face_colored_cube.png?width=300.
- Burnside's_lemma wikiPageID "251900".
- Burnside's_lemma wikiPageRevisionID "604038125".
- Burnside's_lemma hasPhotoCollection Burnside's_lemma.
- Burnside's_lemma subject Category:Group_theory.
- Burnside's_lemma subject Category:Lemmas.
- Burnside's_lemma type Abstraction100002137.
- Burnside's_lemma type Communication100033020.
- Burnside's_lemma type Lemma106751833.
- Burnside's_lemma type Lemmas.
- Burnside's_lemma type Message106598915.
- Burnside's_lemma type Proposition106750804.
- Burnside's_lemma type Statement106722453.
- Burnside's_lemma comment "Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy-Frobenius lemma or the orbit-counting theorem, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. Its various eponyms include William Burnside, George Pólya, Augustin Louis Cauchy, and Ferdinand Georg Frobenius.".
- Burnside's_lemma label "Burnside's lemma".
- Burnside's_lemma label "Lemma van Burnside".
- Burnside's_lemma label "Лемма Бёрнсайда".
- Burnside's_lemma label "伯恩赛德引理".
- Burnside's_lemma sameAs 번사이드_보조정리.
- Burnside's_lemma sameAs Lemma_van_Burnside.
- Burnside's_lemma sameAs m.01lcxr.
- Burnside's_lemma sameAs Q1330377.
- Burnside's_lemma sameAs Q1330377.
- Burnside's_lemma sameAs Burnside's_lemma.
- Burnside's_lemma wasDerivedFrom Burnside's_lemma?oldid=604038125.
- Burnside's_lemma depiction Face_colored_cube.png.
- Burnside's_lemma isPrimaryTopicOf Burnside's_lemma.
