Congruences via Abelian groups

Given a group G acting on a set S, Mobius inversion over the lattice of sub-groups can be used to obtain congruences relating the number of elements of S stabilized by each subgroup. By taking S to be a set of subsets, partitions, or per-mutations congruences for binomial and multinomial coefficients. Stirling numbers of both kinds, and various other combinatorial sequences are derived. Congruences for different moduh are obtained by varying the order of G. m ( 1985 Academic ores, Inc. 1