Co-prime probability for nonabelian metabelian groups of order 24 and their related graphs

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. The study of the co-prime probability and the co-prime graph with different kinds of groups have been widely spread among the researchers for the last few years. Unfortunately, none did a research on both the co-prime probability and its graphs. Hence, this research focuses on the co-prime probability and its graphs for nonabelian metabelian groups of order 24. The research starts by calculating the co-prime probability and later, constructing the co-prime graph. Later, the types of graph, the number of edges, the dominating number and the independent number are obtained. The definitions, theorems and propositions by previous researches are used to achieve the objectives of this research