Burnside's Lemma

Burnside's Lemma as it can be found on MathWorld:

Let J be a finite group and the image R(J) be a representation which is a homomorphism of J into a permutation group S(X), where S(X) is the group of all permutations of a set X. Define the orbits of R(J) as the equivalence classes under x~y ,which is true if there is some permutation p in R(J) such that p(x)=y. Define the fixed points of p as the elements x of X for which p(x)=x. Then the arithmetic mean number of fixed points of permutations in R(J) is equal to the number of orbits of R(J).

Today I deepened my understanding of Burnside's Lemma considerably.