Concept Of Commutativity Degree Research Articles

The Class Equation and the Commutativity Degree for Complete Hypergroups

The aim of this paper is to extend, from group theory to hypergroup theory, the class equation and the concept of commutativity degree. Both of them are studied in depth for complete hypergroups because we want to stress the similarities and the differences with respect to group theory, and the representation theorem of complete hypergroups helps us in this direction. We also find conditions under which the commutativity degree can be expressed by using the class equation.

Bounds on the Action Degree of Groups

The commutativity degree is the probability that a pair of elements chosen randomly from a group commute. The concept of commutativity degree has been widely discussed by several authors in many directions. One of the important generalizations of commutativity degree is the probability that a random element from a finite group G fixes a random element from a non-empty set S that we call the action degree of groups. In this research, the concept of action degree is further studied where some inequalities and bounds on the action degree of finite groups are determined. Moreover, a general relation between the action degree of a finite group G and a subgroup H is provided. Next, the action degree for the direct product of two finite groups is determined. Previously, the action degree was only defined for finite groups, the action degree for finitely generated groups will be defined in this research and some bounds on them are going to be determined.

A GENERALIZATION ON THE NTH COMMUTATIVITY DEGREE OF ALTERNATING GROUPS OF DEGREE 4 AND 5

The theory of commutativity degree is important in determining the abelianness of a group. The commutativity degree of a finite group G is the probability that a pair of elements chosen randomly from a group G, commute. The concept of commutativity degree can be generalized to the nth commutativity degree of a group which is defined as the probability of commuting the nth power of a randomly chosen element with another random element from the same group. In this research, the nth commutativity degree of alternating groups of degree 4 and 5 are presented.