RELATIVE COPRIME PROBABILITY FOR CYCLIC SUBGROUPS OF SOME DIHEDRAL GROUPS

Abstract

A dihedral group is a group of symmetries of a regular -sided polygon, in other words, a structured operation that will make n-gon to go back to itself through a solid motion. Many researchers have studied various fields of group theory using dihedral groups and one of them is the study of the coprime probability of a group and it is defined as the probability of a random pair of elements in a group G such that the greatest common divisor of the order of x and y, where x and y are in G, is equal to one. The coprime probability of G is then extended to the relative coprime  probability G and it is defined as the probability that two randomly selected elements h  from H and g from G where H is a subgroup of a group G such that the greatest common divisor of the order of h and order of g, is equal to one.  In this research, the concentration is on the generalization of the relative coprime probability for cyclic subgroups of some dihedral groups.