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.