Let be a Finite Group and the image be a representation which is a Homeomorphism of into a Permutation Group , where is the Group of all permutations of a Set . Define the orbits of as the equivalence classes under , which is true if there is some permutation in such that . Define the fixed points of as the elements of for which . Then the Average number of Fixed Points of permutations in is equal to the number of orbits of .
The Lemma was apparently known by Cauchy (1845) in obscure form and Frobenius (1887) prior to Burnside's (1900) rediscovery. It was subsequently extended and refined by Pólya (1937) for applications in Combinatorial counting problems. In this form, it is known as Pólya Enumeration Theorem.
Pólya, G. ``Kombinatorische Anzahlbestimmungen für Gruppen, Graphen, und chemische Verbindungen.'' Acta Math. 68, 145-254, 1937.