Solvable not implies nilpotent

From Groupprops
Revision as of 10:38, 8 August 2008 by Vipul (talk | contribs)
Jump to: navigation, search
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

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. Also, for any prime p, the holomorph of the cyclic group of order p (i.e. its semidirect product with its automorphism group) is solvable, but not nilpotent.