Symmetric Perfect and Symmetric Semiperfect Colorings of Groups

Abstract

Let G be a group. A k-coloring of G is a surjection λ:G→{1,2,…,k}. Equivalently, a k-coloring λ of G is a partition P={P1,P2,…,Pk} of G into k subsets. If gP=P for all g in G, we say that λ is perfect. If hP=P only for all h∈H≤G such that [G:H]=2, then λ is semiperfect. If there is an element g∈G such that λ(x)=λ(gx−1g) for all x∈G, then λ is said to be symmetric. In this research, we relate the notion of symmetric colorings with perfect and semiperfect colorings. Specifically, we identify which perfect and semiperfect colorings are symmetric in relation to the subgroups of G that contain the squares of elements in G, in H, and in G∖H. We also show examples of colored planar patterns that represent symmetric perfect and symmetric semiperfect colorings of some groups.