# Abelian 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

A subgroup of a group is termed an **abelian subgroup** if it is abelian as a group.

## Relation with other properties

### Conjunction with other properties

- Central subgroup: An Abelian subgroup that is also a central factor.
- Abelian normal subgroup: An Abelian subgroup that is also a normal subgroup.
- Abelian characteristic subgroup: An Abelian subgroup that is also a characteristic subgroup.
- Maximal among Abelian subgroups: An Abelian subgroup that is also a self-centralizing subgroup.

### Stronger properties

## Metaproperties

### Left-hereditariness

This subgroup property is left-hereditary: any subgroup of a subgroup with this property also has this property. Hence, it is also a transitive subgroup property.

### Join-closedness

This subgroup property is not join-closed, viz., it is not true that a join of subgroups with this property must have this property.

Read an article on methods to prove that a subgroup property is not join-closed

## Effect of property modifiers

### The join-transiter

*Applying the join-transiter to this property gives*: central subgroup