# Group having an abelian normal non-central subgroup

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

A **group having an Abelian normal non-central subgroup** is a group satisfying the following equivalent conditions:

- There exists an Abelian normal subgroup not contained inside the center
- There exists an Abelian normal subgroup properly containing the center

## Relation with other properties

### Stronger properties

- non-Abelian nilpotent group
- non-Abelian supersolvable group
- non-Abelian group in which maximal among Abelian normal implies self-centralizing