Some variations of the commutativity degree of some groups and their applications in graph theory

Abstract

Studying the properties of groups based on some probabilistic methods is an appealing branch of research in group theory. It started by investigating commutativity for symmetric groups, and later grew to a massive number of concepts that measure certain aspects of commutativity for finite and infinite groups. Commutativity degree is defined as the probability that two arbitrary chosen elements of the group commute. This concept has been generalized in several ways, one of these generalizations is the probability of an element of a finite group to fix an element of a finite set, namely the action degree of finite groups. In this thesis, the concept of action degree of finite groups is considered where some inequalities and limiting conditions are determined. Moreover, the definition is extended to the infinite case where the action degree of finitely generated groups is presented along with some bounds of this probability. In a different direction, this research presents a new variation of the commutativity degree of groups, namely the order commutativity degree which is the probability that two elements of the same order of the group commute. This commutativity is proven to be equal to one if and only if the group itself is abelian. The order commutativity degree of finite groups is calculated by dividing the cardinality of the set of all pairs of commutative elements of the same order in the group by the cardinality of the set of all pairs of elements of equal order. To simplify the computations, two formulas are provided to compute the order commutativity degree of finite groups. The first one is by using the number of elements of the same order and the sizes of the centralizers and conjugacy classes for some representative elements, while the second formula depends on a newly defined concept called the order centralizer of elements. Additionally, some explicit formulas are provided to calculate the order commutativity degree for certain types of finite groupswhich are dihedral groups, generalized quaternion groups, groups of composite order, and some groups of prime power order. Later, the order commutativity degree is associated to define a new graph called the order commuting graph in which the vertices of the graph are the elements of a particular order of the group and two vertices are linked by an edge provided that their corresponding elements in the group commute. Moreover, the order commuting graphs are considered and obtained for the previously mentioned groups. These graphs are found to be either empty graphs, complete graphs or bipartite graphs. Finally, several algebraic properties of these order commuting graphs are determined including the degrees of the vertices, graphs independence number, chromatic number, clique number, diameter and girth.