# Group in which every Abelian normal subgroup is central

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

## Contents

## Definition

### Symbol-free definition

A **group in which every Abelian normal subgroup is central** is defined as a group satisfying the following equivalent conditions:

- Any Abelian normal subgroup of the group is a central subgroup, i.e., is contained in the center.
- The center is maximal among Abelian normal subgroups.

## Relation with other properties

### Stronger properties

### Weaker properties

### Opposite properties

- Group in which maximal among Abelian normal implies self-centralizing: The only groups with
*both*these properties are the Abelian groups.