Coprime Degree of Some Groups and Its Relation to Coprime Graphs
Abdul Gazir Syarifudin (a), Rian Kurnia (a), Pudji Astuti Waluyo (a), Ahmad Muchlis (a*)

(a) Faculty of Mathematics and Natural Sciences, Bandung Institute of Technology, 40132 Bandung, Indonesia
*muchlis[at]math.itb.ac.id

Abstract

Coprime degree of a group can be thought of as a generalization of commutativity degree of a group. Coprime degree of a group is defined to be the probability that two random elements in the group have coprime orders. In this paper, we derive explicit description of coprime degree for several finite groups with respect to the prime decompositions of their group orders, including cyclic groups, dihedral groups, and generalized quaternion groups. Furthermore, we establish the relation between coprime degree of a group and its coprime graph.

Keywords: Coprime degree, Commutativity degree, Coprime graph, Cyclic groups, Dihedral groups, Generalized quaternion groups.

Topic: MATHEMATICS AND STATISTICS