Many years ago this problem was solved for solvable groups.

The fact plays a crucial role in the structure theory of solvable groups.

In fact, they are precisely the solvable groups of derived length at most 2.

The terminology arises from the solvable groups of abstract group theory.

Solvable groups are too messy to classify except in a few small dimensions.

These can be particularly useful in the study of solvable groups and nilpotent groups.

Furthermore, it follows that all solvable groups are amenable.

Commutators are used to define nilpotent and solvable groups.

By contrast, for a solvable group the definition requires each quotient to be abelian.

A solvable group is one whose derived series reaches the trivial subgroup at a 'finite' stage.

Shafarevich showed that every finite solvable group is realizable over 'Q'.

Clearly all solvable groups are virtually solvable, since one can just choose the group itself, which has index 1.

Every nilpotent group, and more generally, every solvable group, is thin.

His major work was on group theory, notably on finite groups and solvable groups.

This is particularly relevant to the representation theory of finite solvable groups, where normal subgroups usually abound.

More generally, every residually solvable group is hypoabelian.

Every periodic solvable group is locally finite .

As every finite solvable group is polycyclic, this can be seen as one of the key differences between the definitions.

Every solvable group is hypoabelian, and so is every free group.

A solvable group, or soluble group, is one with a subnormal series whose factor groups are all abelian.

In finite solvable groups, every polynormal subgroup is paranormal.

Thus a finite almost simple group is an extension of a solvable group by a simple group.

Every metanilpotent group is a solvable group.

The alternating group 'A'4 is an example of a finite solvable group that is not supersolvable.

Lattice-theoretic characterizations of this type also exist for solvable groups and perfect groups .