The Quartic Commutativity Degree of Dihedral Groups

Abstract

The combination of group theory and probability theory was used in studying the connection between the two. In recent years, probability theory has been widely used in solving several difficult problems in group theory. The commutativity degree of a group, is defined as the probability that two random elements in a group commute. In addition, there exist a generalization of commutativity degree of a group which is the -th power commutativity degree of a group and it is defined as the probability of the -th power of two random elements in a group commute. Some previous studies have been found for equal to 2 and 3 and both probabilities are called as squared commutativity degree and cubed commutativity degree respectively. In this research, the -th power commutativity degree is determined for equal to 4, called as quartic commutativity degree and some generalization formulas have been obtained. However, this research focuses only on the dihedral groups.