Point groups of order at most eight and their conjugacy class graphs

Abstract

Point group is a type of group in chemistry, which is a collection of symmetry elements possessed by a shape or form which all pass through one point in space. The stereographic projection is used to visualize the symmetry operations of point groups. On the other hand, group theory is the study about an algebraic structure known as a group in mathematics. This research relates point groups of order at most eight with groups in group theory. In this research, isomorphisms, matrix representations, conjugacy classes and conjugacy class graph of point groups of order at most eight are found. The isomorphism between point groups and groups in group theory are obtained by mapping the elements of the groups and by showing that the isomorphism properties are fulfilled. Then, matrix representations of point groups are found based on the multiplication table. The conjugacy classes and conjugacy class graph of point groups of order at most eight are then obtained. From this research, it is shown that point groups C1, C3, C5 and C7 are isomorphic to the groups Z1, Z3, Z5 and Z7 respectively. Next, point groups C1h = Cs = C1v = S1, S2 = Ci, C2 = D1 are isomorphic to the group Z2, point groups C4, S4 are isomorphic to the group Z4, point groups C2h = D1d, C2v = D1h, D2 are isomorphic to the group Z2 × Z2, point groups C3v, D3 are isomorphic to the group S3, point groups C6, S6, C3h = S3 are isomorphic to the group Z6. The conjugacy classes of these groups are then applied to graph theory. It is found that the conjugacy class graph for point groups C3v and D3 are empty graphs; while conjugacy class graph for point groups C4v, D4 and D2d are complete graphs. As a conclusion, point groups in chemistry can be related with group theory in term of isomorphisms, matrix representations, conjugacy classes and conjugacy class graph.