Solvable not implies nilpotent

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., solvable group) need not satisfy the second group property (i.e., nilpotent group)
View a complete list of group property non-implications | View a complete list of group property implications

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.

Related facts

Converse

The converse is true: nilpotent implies solvable.

Sources of examples

Note that prime power order implies nilpotent, so any finite example must be of order not a prime power order. Further, equivalence of definitions of finite nilpotent group tells us that any nilpotent group must be the direct product of its Sylow subgroups. Hence, we must look for groups whose order is not a prime power, and that are not direct products of their Sylow subgroups, or equivalently, that have non-normal Sylow subgroups.

We also need to make sure that the group remains solvable. It turns out that order has only two prime factors implies solvable, so any non-nilpotent group whose order has only two prime factors must give an example.

Further, since nilpotency is subgroup-closed and solvability is finite direct product-closed, we can take the direct product of any non-nilpotent solvable group with any (nilpotent or not) solvable group and get more examples of non-nilpotent solvable groups.

Related specific information

Numerical information on number of nilpotent and solvable groups for orders that have only two prime factors

Order Prime factors Information on groups of this order Number of nilpotent groups Number of solvable groups Comment
6 2, 3 groups of order 6 1 2 symmetric group:S3 is the non-nilpotent solvable group.
10 2, 5 groups of order 10 1 2 dihedral group:D10 is the non-nilpotent solvable group.
12 2, 3 groups of order 12 2 5 alternating group:A4, dihedral group:D12, and dicyclic group:Dic12 are the non-nilpotent solvable groups.
14 2, 7 groups of order 14 1 2 dihedral group:D14 is the non-nilpotent solvable group.
18 2,3 groups of order 18 2 5 dihedral group:D18, direct product of S3 and Z3, and generalized dihedral group for E9 are the non-nilpotent solvable groups.
20 2, 5 groups of order 20 2 5 dihedral group:D20, dicyclic group:Dic20, and general affine group:GA(1,5) are the non-nilpotent solvable groups.
24 2, 3 groups of order 24 5 15 Follow link to get list
48 2, 3 groups of order 48 14 52 Follow link to get list