The conjugacy classes and related graphs of some three-generator p-groups

Abstract

A conjugacy class is a set of elements in the group under the conjugation action. Meanwhile, a graph consists of points called vertices, and links which are called edges, is indicated by line segments or curves joining certain pairs of vertices. In the first part of this research, the conjugacy class of each type of groups in the classification is computed based on the definition of conjugacy class itself. Previous researchers have found the lower and upper bound of conjugacy classes. Compared to those researches, this research move a step further where the exact number of conjugacy classes of groups are computed. These results are then applied into finding its conjugacy class graph. In the second part of this research, the conjugacy classes of the groups satisfying the set omega is determined. A set omega is defined as the set of all ordered pairs of commuting elements in the classification. In this research, the classification of 3-generator p-groups of certain order is used. The conjugacy classes is computed by using several steps. Firstly, the elements of order p is identified, followed by the determination of the number of elements of ordered pairs in the set omega. The number of conjugacy classes is then obtained and expressed in a general form. The obtained results are later associated to two types of graph, namely the orbit graph and the generalized conjugacy class graph. For the orbit graph, two vertices are adjacent if they are conjugated to each other. Meanwhile, for the generalized conjugacy class graph, two vertices are adjacent to each other if their cardinalities are not coprime. The graphs turned out to be a complete graph or a union of complete graphs with p vertices. In the last part of this research, two new graphs of 3-generator 3-groups called the generalized commuting conjugacy class graph and the generalized non-commuting conjugacy class graph are introduced. The generalized commuting conjugacy class graph is a graph whose the vertices are the elements of the conjugacy classes in the form of ordered pairs. Two vertices are adjacent if they are commuted. Since the generalized non-commuting conjugacy class graph is a complement of the generalized commuting conjugacy class graph, thus, the edges are connected if their vertices are not commuted.