Hyperabelian group

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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.
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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



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