Generalized commutativity degrees of some finite groups and their related graphs

Abstract

The commutativity degree of a finite group G is the probability that two randomly chosen elements of the group G commute and is denoted as P(G). The concept of commutativity degree is then extended to the relative commutativity degree of a groupG, denoted as P(H,G), which is defined as the probability that two arbitrary elements, one in the subgroup H and another in the group G, commute. Similarly, the concept of commutativity degree can be extended to two arbitrary elements from two subgroups of the group. In this research, the theory of commutativity degree is extended by defining the probability that the n-th power of a random pair of elements in the group G commute, where it is called the n-th power commutativity degree and is denoted by Pn(G). The computation of n-th power commutativity degree are divided into two cases, namely for n = 2 and n = 3 where P2(G) and P3(G) is called the squared commutativity degree and cubed commutativity degree, respectively. These probabilities have been obtained for dihedral groups. Meanwhile, the productivity degree of two subgroups of a group is also an extension of the commutativity degree and it is defined as the ratio of the order of the intersection of HK and KH with the order of their union, where H and K are two subgroups of a group G, denoted by PG(HK). The general formula for dihedral groups has been found for this probability. Another extension of the commutativity degree which has been defined in this research is the relative n-th nilpotency degree of two subgroups of a group, denoted as Pnil(n,H,K). This probability is defined as the probability that the commutator of two arbitrary elements in H and in K belongs to the n-th central series of the group. Some results that have been found through this probability include its lower and upper bound, its comparison between their factor groups and its relation with extra relative n-isoclinism. All of the results obtained are then applied to graph theory where a graph related to each probability is defined. This includes the graph related to cubed commutativity degree and the product of subgroup graph which is related to the productivity degree of two subgroups. A new graph which is called a complete tripartite graph is introduced. The last graph is the bipartite graph associated to a non-nilpotent group of class (n - 1), called as relative non-nil (n - 1) bipartite graph. Therefore, some graph properties are found for all mentioned graphs which include the diameter and girth.