Finite Groups with Non-Commuting Graphs

Group  is called metacyclic if it contains a normal cyclic subgroup such that the quotient group  is also cyclic. In this paper, two conjectures proposed by Abdollahi et al. (2006) for a family of finite non-abelian metacyclic prime power groups  were investigated. For this purpose, first, the metacyclic groups were categorized into three Types (families) of the non-isomorphic groups. Next, by using the size of centralizers and also equality of the conjugacy vector type ctv (G) of these groups, the necessary and sufficient conditions under which two non-abelian finite metacyclic prime power groups have the isomorphic non-commuting graphs were determined. The first conjecture of Abdollahi et al. for the three families of the classified groups was proven to be true. Likewise, the second conjecture held for some restrictions on the parameters of group . Finally, it was demonstrated that there were non-isomorphic groups with the same non-commuting graphs.