# Abelian hereditarily normal subgroup

From Groupprops

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

This article describes a property that arises as the conjunction of a subgroup property: hereditarily normal subgroup with a group property (itself viewed as a subgroup property): abelian group

View a complete list of such conjunctions

This article describes a property that arises as the conjunction of a subgroup property: transitively normal subgroup with a group property (itself viewed as a subgroup property): abelian group

View a complete list of such conjunctions

## Definition

A subgroup of a group is termed an **abelian hereditarily normal subgroup** or **abelian transitively normal subgroup** of if it satisfies the following equivalent conditions:

- is abelian as a group and is a hereditarily normal subgroup of .
- is abelian as a group and is a transitively normal subgroup of .
- is an abelian normal subgroup of and the induced action of the quotient group on is by power automorphisms.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Central subgroup | ||||

Cyclic normal subgroup |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Hereditarily normal subgroup | ||||

Transitively normal subgroup |