# Hypoabelian 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 is a variation of solvable group|Find other variations of solvable group |

This is an opposite of perfect group

## Definition

### Symbol-free definition

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

1. The perfect core is trivial
2. The hypoabelianization is the quotient by the trivial subgroup, and hence, isomorphic to the whole group.
3. 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.)
4. There is no nontrivial perfect subgroup.
5. 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