The conjugacy classes and conjugacy class graphs of point groups of order at most 8

Abstract

In group theory, conjugacy class is a method of partitioning the elements of a group such that the elements a and b are conjugate in a group G if xax−1 = b for some x in G. Meanwhile, a point group is a set of symmetry operations that keeps at least one point in a molecule fixed. In chemistry, symmetry of molecules is important since chemists classify molecules based on their symmetry. In this research, the conjugacy classes of point groups of order at most eight are computed. The conjugacy classes of these groups are then applied to graph theory to obtain the conjugacy class graph, which is a graph whose vertices V = (v1,…,vn) are the non-central conjugacy classes of a group and the two vertices are connected if their cardinalities are not coprime, that is the greatest common divisor of the cardinalities of the corresponding vertices is not equal to one.