A generalization of Burnside's combinatorial lemma

Abstract

Let G be a finite group which acts on a set S. We present a method of computing the entire distribution of G-orbits of S (the number of k-element G-orbits of S for all k) in terms of the number of s ϵ S fixed by every σ ϵ H for subgroups H of G, and the Möbius function μ(·, ·) defined on the subgroup lattice of G. We deduce Burnside's lemma as a consequence of our result.