Let 
be a finite group and the image 
be a representation which is a homomorphism
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 arithmetic mean 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 is sometimes also called Burnside's lemma, the orbit-counting theorem, the Pólya-Burnside lemma, or even "the lemma that is not Burnside's!" Whatever its name, the lemma was subsequently extended and refined by Pólya (1937) for applications in combinatorial counting problems. In this form, it is known as Pólya enumeration theorem.
."
Amer. Math. Monthly 106, 525-533, 1999.Pólya, G.
"Kombinatorische Anzahlbestimmungen für Gruppen, Graphen, und chemische
Verbindungen." Acta Math. 68, 145-254, 1937.Rotman,
J.