Solvable not implies nilpotent
From Groupprops
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
Get more facts about solvable group|Get more facts about nilpotent group
Contents
Statement
Not every solvable group is nilpotent.
Definitions used
Term | Definition used |
---|---|
nilpotent group | The upper central series terminates at the whole group. For a finite group, this is equivalent to being the direct product of its Sylow subgroups (see finite nilpotent group, equivalence of definitions of finite nilpotent group |
solvable group | There is a subnormal series where all the successive quotient groups are abelian groups. |
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 , the general affine group of degree one , which can also be defined as the holomorph of the cyclic group of order (i.e. its semidirect product with its automorphism group) is solvable, but not nilpotent.
- if and are primes such that divides , there is a solvable non-nilpotent group of order . See classification of groups of order pq.
Related facts
Converse
The converse is true: nilpotent implies solvable.
Fact | Nature of significance | Details |
---|---|---|
prime power order implies nilpotent | where not to look if you want to avoid nilpotent | to make sure the group is non-nilpotent, do not look at prime powers. They will not work. |
equivalence of definitions of finite nilpotent group | where to look and where not to look if you want to avoid nilpotent | any nilpotent group must be the direct product of its Sylow subgroups, or equivalently, all its Sylow subgroups are normal. Hence, we must look for groups that are not direct products of their Sylow subgroups, or equivalently, that have non-normal Sylow subgroups. |
order has only two prime factors implies solvable | where to look if you want to ensure solvable | any group whose order is of the form is automatically solvable, so as long as we make sure that the group isn't nilpotent, we have an example. |
nilpotent of cube-free order implies abelian | where to look if you want to avoid nilpotent | any example of a solvable non-abelian group where the order is a cube-free number automatically gives an example of a solvable non-nilpotent group. This is because if the group were nilpotent, it would be abelian on account of its cube-free order. |
nilpotency is subgroup-closed, nilpotency is quotient-closed, solvability is finite direct product-closed, solvability is extension-closed | how to construct bigger examples from smaller | if is a non-nilpotent solvable group, and is any solvable group (possibly nilpotent, possibly not nilpotent), is also a non-nilpotent solvable group. More generally, any extension with normal subgroup and quotient group , or with normal subgroup and quotient , is a non-nilpotent solvable group. |
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 |