# 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:

- 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.
- The group is either Abelian or it is a one-headed group where the head is the center.
- Every proper subnormal subgroup is a central subgroup.