Supersolvable not implies nilpotent

Statement
It is possible for a group to be a supersolvable group but not a nilpotent group.

Converse
We have finite nilpotent implies supersolvable. Note that infinite nilpotent groups, and even infinite abelian groups, need not be abelian, because of finiteness problems. For instance, the group of rational numbers (additive) is not supersolvable.

Similar facts

 * Solvable not implies supersolvable

Opposite facts

 * Supersolvable implies nilpotent derived subgroup

Proof
The symmetric group of degree three is an example of a finite supersolvable group that is not nilpotent:


 * The group has a normal subgroup A3 in S3 such that both the subgroup and the quotient are cyclic groups, so it is supersolvable.
 * The group is centerless, so it is not nilpotent.