Groups with few non-normal cyclic subgroups

Abstract

R. Brandl in [2] and H. Mousavi in [6] classified finite groups which have respectively just one or exactly two conjugacy classes of non-normal subgroups. In this paper we determine finite groups which have just one or exactly two conjugacy classes of non-normal cyclic subgropus. In particular, in a nilpotent group if all non-normal cyclic subgroups are conjugate, then any two non-normal subgroups are conjugate. In general, if a group has exactly two conjugacy classes of non-normal cyclic subgroups, there is no upper bound for the number of conjugacy classes of non-normal subgroups.