# 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 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