Solvable not implies nilpotent
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Revision as of 20:16, 7 March 2008 by Vipul (talk  contribs) (New page: {{group property nonimplication}} ==Statement== Not every solvable group is nilpotent. ==Proof== The smallest solvable nonnilpotent group is [[symmetric group...)
This article gives the statement and possibly, proof, of a nonimplication relation between two group properties. That is, it states that every group satisfying the first group property need not satisfy the second group property
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Statement
Not every solvable group is nilpotent.
Proof
The smallest solvable nonnilpotent 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 , the holomorph of the cyclic group of order (i.e. its semidirect product with its automorphism group) is solvable, but not nilpotent.