Co-prime and relative co-prime probability for nonabelian metabelian groups of order at most 24 with their related graphs

Abstract

The concept of probability involving groups started with a notion known as the commutativity degree of a group. Later, a new definition is introduced namely the co-prime probability of a group. The probability of a random pair of elements in a group G is said to be co-prime when the greatest common divisor of the order of x and y, where x and y are in G, is equal to one. This co-prime probability is then further extended to the relative co-prime probability of a group and it is newly defined in this dissertation. The probability that two randomly selected elements from H and G is called relative co-prime when the greatest common divisor of the order of h and g, where h is in H and g is in G, is equal to one. This dissertation also discusses on the co-prime graph whereby 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 graph is then extended to the relative coprime graph where the vertices are elements of a group and two distinct vertices are adjacent if and only if their orders are co-prime and any of them is in the subgroup of the group. Past researchers studied the co-prime probability and their related graphs as well as relative co-prime graph on p-groups and dihedral groups but none did on the nonabelian metabelian groups. Hence, this dissertation aims to be more specific by determining both the co-prime and relative co-prime probability together with their related graphs for nonabelian metabelian groups of order at most 24. The number of edges, the types of the graph and the properties of the graph such as the dominating number and the independent number are discussed. Both Maple 2016 software and some related results by previous researches are used in order to achieve the objectives of this dissertation. It is found that for the co-prime and relative co-prime probability for nonabelian metabelian groups of order at most 24, the results varies for each group with different order. As for the co-prime and relative co-prime graph, the results shows that the dominating number for each group is one while the number of edges, the types of graph and the independent number for each group varies with different order.