# Hypoabelian 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

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 solvable group|Find other variations of solvable group |

*This is an opposite of perfect group*

## 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 quotient by the trivial subgroup, and hence, isomorphic to the whole group.
- The transfinite derived series terminates at the identity. (Note that this is the
*transfinite*derived series, where the successor of a given subgroup is its commutator subgroup and subgroups at limit ordinals are given by intersecting all previous subgroups.) - There is no nontrivial perfect subgroup.
- There is a descending transfinite normal series where all the successive quotients are abelian

## Relation with other properties

### Stronger properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Solvable group | derived series terminates at identity in finitely many steps | solvable implies hypoabelian | hypoabelian not implies solvable | Residually solvable group|FULL LIST, MORE INFO |

Hypocentral group | transfinite lower central series terminates at identity | hypocentral implies hypoabelian | hypoabelian not implies hypocentral | |FULL LIST, MORE INFO |

Residually solvable group | intersection of all finite members of derived series is identity | residually solvable implies hypoabelian | hypoabelian not implies residually solvable | |FULL LIST, MORE INFO |

Free group | Free on a given generating set | (via residually solvable) | (via residually solvable) | Residually nilpotent group, Residually solvable group|FULL LIST, MORE INFO |