On the orbits of some metabelian groups

Abstract

In this study, the orbits of non abelian metabelian groups of order 26, 28 and 30 are found using conjugation action. The non abelian metabelian groups considered in this study are some dihedral groups D₁₃, D₁₄ and D₁₅ as well as the semidirect products D₇ × Z₂, D₅ × Z₃ and D₃ × Z₅. In order to obtain the orbit, the set of all pairs of commuting elements of the groups of size two is found. The results obtained from the orbits are then applied into an extension of commutativity degree which is the probability that an element of the groups fixes a set. Finally, the generalized conjugacy class graph is constructed and its chromatic number and clique number are also found.