Group in which every proper normal subgroup is central

Definition
A group in which every proper normal subgroup is central is a group satisfying the following equivalent conditions:


 * 1) Every proper normal subgroup (i.e., every normal subgroup other than the whole group) of the group is a central subgroup: it is contained in the center.
 * 2) The group is either Abelian or it is a one-headed group where the head is the center.
 * 3) Every proper subnormal subgroup is a central subgroup.