Some characterizations of groups of order 8

Abstract

Group theory is a branch of mathematics which concerns with the study of groups. It has wide applications in other fields too including chemistry. This research focuses on groups of order 8 and their irreducible representations. There are five groups of order 8, namely 0 4 , Q, C8, C2 x C4 and C2 x C2 x C2. For any group, the number of possible representative sets of matrices is infinite, but they can all be reduced to a single fundamental set, called the irreducible representations of the group. Burnside method and Great Orthogonality Theorem method are both used to obtain irreducible representations of all groups of order 8. Then, comparisons of both methods are made. Irreducible representation is actually the nucleus of a character table and is of great importance in chemistry. Groups of order 8 are isomorphic to certain point groups. Point groups are symmetry groups which leave at least one point in space fixed under all operations. In this research, isomorphisms from four out of five groups of order 8, namely 0 4 , C8, C2 x C4 and C2 x C2 x C2, and isomorphisms from proper subgroups of Q to certain point groups are determined.