Burnside’s Lemma and a Few of Its Applications

The action of a finite group G on a given nonempty set X produces a partition of X into mutually disjoint classes, called the orbits of G. A formula for the number of orbits of G is derived. It has various applications: one is a number-theoretic identity (due to P. Kesava Menon) which comes out as a direct application of Burnside’s lemma to the set of residue classes (mod r), under the action of the group of reduced residue classes (mod r).