Burnside's Lemma states that the number of distinct configurations of a set under the action of a symmetry group equals the average, taken over every element of that group, of the number of configurations each element leaves fixed. Named for William Burnside, who popularized it though it was known earlier to Augustin-Louis Cauchy and Ferdinand Georg Frobenius, it is the standard tool for counting objects up to symmetry, such as distinct necklaces or dice colorings, and underlies the more general Polya Enumeration Theorem.
Facts
StatementBurnside's lemma states that the number of distinct configurations of a set under the action of a finite group equals the average, taken over every element of the group, of the number of configurations that element leaves fixed. 1 Proof YearYear of Burnside's own book, the source of the entity's name; the same result was stated and proved earlier by Frobenius in 1887, itself after Cauchy in 1845, a Stigler's law of eponymy case. Classification
Statement Form Connections
Sources
1. Burnside's Lemma (Wikipedia)
Wikimedia Foundationlead paragraph, attribution sentence
It was discovered by Augustin Louis Cauchy and Ferdinand Georg Frobenius, and became well known after William Burnside quoted it.
History: the lemma that is not Burnside's section
William Burnside stated and proved this lemma in his 1897 book on finite groups, attributing it to Frobenius 1887. But even prior to Frobenius, the formula was known to Cauchy in 1845.
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