Co-prime Probability for Nonabelian Metabelian Groups of Order Less than 24 and Their Related Graphs

Abstract

Let G be a finite group. The probability of a random pair of elements in G are said to be co-prime when the greatest common divisor of order x and y, where x and y in G, is equal to one. Meanwhile the co-prime graph of a group is defined as a graph whose vertices are elements of G and two distinct vertices are adjacent if and only if the greatest common divisor of order x and y is equal to one. In this paper, the co-prime probability and its graph such as the type and the properties of the graph are determined.