Related graphs of the conjugacy classes of a 3-generator 5-group

Abstract

The study on conjugacy class has started since 1968. A conjugacy class is defined as an equivalence class under the equivalence relation of being conjugate. In this research, let be a 3-generator 5-group and the scope of the graphs is a simple undirected graph. This paper focuses on the determination of the conjugacy classes of where the set omega is the subset of all commuting elements in the group. The elements of the group with order 5 are identified from the group presentation. The pair of elements are formed in the form of  which is of size two where  and  commute. In addition, the results on conjugacy classes of are applied into graph theory. The determination of the set omega is important in the computation of conjugacy classes in order to find the generalized conjugacy class graph and orbit graph. The group action that is considered to compute the conjugacy classes is conjugation action. Based on the computation, the generalized conjugacy class graph and orbit graph turned out to be a complete graph.