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Solvable not implies nilpotent

2,520 bytes added, 21:36, 18 June 2011
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The smallest solvable non-nilpotent group is [[symmetric group:S3|the symmetric group on three letters]]. This is [[centerless group|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 <math>p</math>, the [[general affine group of degree one]] <math>GA(1,p)</math>, which can also be defined as the [[holomorph]] of the cyclic group of order <math>p</math> (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=== {| class="sortable" border="1"! 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|}
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