# Abelian-quotient subgroup

From Groupprops

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is termed an **Abelian-quotient subgroup** if it satisfies the following equivalent conditions:

- It contains the commutator subgroup
- It is a normal subgroup and the quotient of the whole group by it is Abelian

### Definition with symbols

A subgroup of a group <math>G</math> is termed an **Abelian-quotient subgroup** if it satisfies the following equivalent conditions:

- where </math>G' = [G,G]</math> is the commutator subgroup of
- is normal in and is an Abelian group

## Relation with other properties

### Stronger properties

- Elementary Abelian-quotient subgroup
- Cyclic quotient-group
- Abelian-completed normal subgroup
- Cocentral subgroup