# Subgroup of abelian normal subgroup

From Groupprops

Revision as of 11:51, 27 March 2009 by Vipul (talk | contribs) (moved Subgroup of Abelian normal subgroup to Subgroup of abelian normal subgroup)

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: subgroup and Abelian normal subgroup

View other such compositions|View all subgroup properties

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: normal subgroup and Abelian normal subgroup

View other such compositions|View all subgroup properties

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: central subgroup and normal subgroup

View other such compositions|View all subgroup properties

## Contents

## Definition

### Symbol-free definition

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

- It is a subgroup of an Abelian normal subgroup.
- It is a normal subgroup of an Abelian normal subgroup.
- It is a central subgroup (i.e., it is contained in the center) of a normal subgroup.

### Equivalence of definitions

`Further information: equivalence of definitions of subgroup of Abelian normal subgroup`

## Relation with other properties

### Stronger properties

- Central subgroup
- Abelian normal subgroup
- 2-subnormal subgroup of least prime order:
`For full proof, refer: 2-subnormal of least prime order implies subgroup of Abelian normal subgroup`