# Difference between revisions of "Hypoabelian group"

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* The [[perfect core]] is [[trivial group|trivial]] | * The [[perfect core]] is [[trivial group|trivial]] | ||

* The [[hypoabelianization]] is the whole group | * The [[hypoabelianization]] is the whole group | ||

− | * The [[derived series]] terminates at the identity | + | * The transfinite [[derived series]] terminates at the identity |

* There is no nontrivial [[perfect group|perfect]] subgroup. | * There is no nontrivial [[perfect group|perfect]] subgroup. | ||

− | * There is a descending [[normal series]] where all the successive quotients are abelian | + | * There is a descending transfinite [[normal series]] where all the successive quotients are abelian |

==Relation with other properties== | ==Relation with other properties== |

## Revision as of 14:24, 24 October 2009

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

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This is a variation of solvability|Find other variations of solvability |

*This is an opposite of perfectness*

## Contents

## Definition

### Symbol-free definition

A group is termed **hypoabelian** if the following equivalent conditions are satisfied:

- The perfect core is trivial
- The hypoabelianization is the whole group
- The transfinite derived series terminates at the identity
- There is no nontrivial perfect subgroup.
- There is a descending transfinite normal series where all the successive quotients are abelian