# Group with nilpotent derived 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

## Contents

## Definition

A **group with nilpotent commutator subgroup**, also called a **nilpotent-by-abelian group**, is a group satisfying the following equivalent conditions:

- Its Commutator subgroup (?) is a Nilpotent group (?).
- It has a Nilpotent normal subgroup (?) with an abelian quotient group.
- It has a Nilpotent characteristic subgroup (?) with an abelian quotient group.

## Relation with other properties

### Stronger properties

### Weaker properties

- Group satisfying subnormal join property:
*For proof of the implication, refer Nilpotent commutator subgroup implies subnormal join property and for proof of its strictness (i.e. the reverse implication being false) refer Subnormal join property not implies nilpotent commutator subgroup*.