Symmetric Perfect and Symmetric Semiperfect Colorings of Groups

Let G be a group. A k-coloring of G is a surjection lambda : G -> {1, 2,..., k }. Equivalently, a k-coloring lambda of G is a partition P = {P1, P2,..., P} g of G into k subsets. If gP - P for all g in G, we say that lambda is perfect. If hP = P only for all h is an element of H <= G such that [G : H] = 2, then lambda is semiperfect. If there is an element g is an element of G such that lambda(x) = lambda l( gx(- 1)g) for all x is an element of G, then l 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.

Automated summary

In this research, the notion of symmetric colorings is related to perfect and semiperfect colorings. Specifically, we identify the symmetry in perfect and semiperfect colorings in relation to subgroups containing elements' squares in G, in H, and in G\H. Examples of colored planar patterns representing symmetric perfect and symmetric semiperfect colorings of some groups are also shown.