NAC-Groups

A finite non-Dedekind group G is called an NAC-group if all non-normal abelian subgroups are cyclic. In this paper, we classify all finite NAC-groups. We show that the center of such groups is cyclic. If G has a non-abelian non-normal Sylow subgroup of odd order, then other Sylow subgroups of G are cyclic or of Quaternion type.