Computing the number of symmetric colorings of elementary Abelian groups

Abstract

Given a finite group G and a positive integer r, an r-coloring of G is any mapping χ:G→{1,…,r}. Colorings χ and φ are equivalent if there exists g∈G such that χ(xg-1)=φ(x) for all x∈G. A coloring χ is symmetric if there exists g∈G such that χ(gx-1g)=χ(x) for every x∈G. We compute the number of symmetric r-colorings and the number of equivalence classes of symmetric r-colorings of an elementary Abelian p-group.