# Group whose center is a direct factor

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

A **group whose center is a direct factor** is a group satisfying the following equivalent conditions:

- It is the direct product of a centerless group and an abelian group.
- Its center is a permutably complemented subgroup.
- Its center is a complemented normal subgroup.
- Its center is a direct factor.

## Relation with other properties

### Stronger properties

- Abelian group
- Centerless group
- Simple group
- Characteristically simple group
- Group in which every normal subgroup is a direct factor

### Weaker properties

- Group whose center is an AEP-subgroup
- Group having an automorphism whose restriction to the center is the inverse map

### Opposite properties

- Group whose center is normality-large
- (non-abelian) nilpotent group