# Hyperabelian group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

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This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to Hyperabelian group, all facts related to Hyperabelian group) |Survey articles about this | Survey articles about definitions built on this

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## Definition

### Symbol-free definition

A **hyperabelian group** is a group which possesses an ascending (possibly transfinite) normal series where all the successive quotients are Abelian.

## Relation with other properties

### Stronger properties

### Weaker properties

## Metaproperties

### Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property

View a complete list of subgroup-closed group properties

The ascending normal series for the subgroup is simply the intersection with the subgroup of the ascending normal series for the whole group.

The ascending normal series for the quotient is the image of the ascending normal series for the whole group via the quotient map.

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