On the orbit of some metabelian groups of order 24 and its applications

An orbit is defined as the partition of an equivalent relation of elements in a group. In order to obtain the orbit, a group action acting on the elements of the groups is considered. In this study, the orbits of some metabelian groups are found using conjugation action. The metabelian groups considered in this study are some nonabelian metabelian groups of order 24, which are the dihedral group, D12 as well as the semidirect products, R = ℤ3 ⋊ ℤ8 and S = ℤ3 ⋊ Q. The results obtained from the orbits are then applied into an extension of commutativity degree, which is the probability that a group element fixes a set where the probability is the ration of the number of orbits and the number of elements in the set. The set considered in this study is the set of all pairs of commuting elements of the groups that is in the form of (x, y), where lcm(|x|, |y|) = 2. Then, the results of the orbits will also be applied into graph theory, specifically generalized conjugacy class graph where its vertices are the non-central orbits and two vertices are adjacent if the cardinality of the orbits is not coprime. Lastly, some properties of the graph which are the chromatic number and clique number are obtained.