Hypoabelian group

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 derived 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	Meaning	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	|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)	|FULL LIST, MORE INFO