Symmetric colorings of finite groups

Abstract

Let G be a finite group and let r ∈ N. A coloring of G is any mapping : G −→ {1, 2, 3, ..., r}. Colorings of G, and are equivalent if there exists an element g in G such that (xg−1) = (x) for all x in G. A coloring of a finite group G is called symmetric with respect to an element g in G if (gx−1g) = (x) for all x ∈ G. We derive formulae for computing the number of symmetric colorings and the number of equivalence classes of symmetric colorings for some classes of finite groups