Solvable not implies nilpotent

Statement
Not every solvable group is nilpotent.

Proof
The smallest solvable non-nilpotent group is the symmetric group on three letters. This is centerless, so it cannot be nilpotent. On the other hand, it is clearly solvable, because its commutator subgroup is the alternating group on three letters, which is Abelian.

More generally:


 * any dihedral group whose order is not a power of 2, is solvable but not nilpotent.
 * for any prime $$p$$, the general affine group of degree one $$GA(1,p)$$, which can also be defined as the holomorph of the cyclic group of order $$p$$ (i.e. its semidirect product with its automorphism group) is solvable, but not nilpotent.
 * if $$p$$ and $$q$$ are primes such that $$p$$ divides $$q - 1$$, there is a solvable non-nilpotent group of order $$pq$$. See classification of groups of order pq.

Converse
The converse is true: nilpotent implies solvable.