The n-th coprime probability and its graph for some dihedral groups

Abstract

The coprime probability and graph have been studied for various groups by many researchers focusing on the generalization of the probability part. For the coprime graph, the types and properties of the graph have been investigated and the patterns that can be found within a group are analysed. The coprime probability of a group is defined as the probability that the order of a random pair of elements in the group are relatively prime or coprime. Meanwhile, the coprime graph can be explained as a graph whose vertices are elements of a group and two distinct vertices are adjacent if and only if the greatest common divisor of the order of the first vertex and order of the second vertex is equal to one. It was unfortunate that the exploration of probabilities and graphs of groups have not considered both the n-th coprime probability and its graph that ultimately became the target in this research. Hence, the newly defined terms are then used to find the generalizations of the n-th coprime probability and the n-th coprime graphs for some dihedral groups. The types and properties of the graphs are also discussed in this research.