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Solvable not implies nilpotent

1,034 bytes added, 21:47, 18 June 2011
Related facts
The converse is true: [[nilpotent implies solvable]].
===Sources of Observations related to search for examples===
Note that {| class="sortable" border="1"! 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, so any finite example must be of order do ''not a '' look at prime power orderpowers. They will not work. Further, |-| [[equivalence of definitions of finite nilpotent group]] tells us that || 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 subgroup]]s, or equivalently, all its Sylow subgroupsare [[normal subgroup|normal]]. Hence, we must look for groups whose order is not a prime power, and that wthat 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]]|| where to look if you want to ensure solvable || any group whose order is of the form <math>p^aq^b</math> 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-''anyabelian'' group where the order is a cube-free number automatically gives an example of a solvable non-nilpotent group whose . This is because if the group were nilpotent, it would be abelian on account of its cube-free order has only two prime factors must give an example.|-Further| [[nilpotency is subgroup-closed]], since [[nilpotency is subgroupquotient-closed]] and , [[solvability is finite direct product-closed]], we can take the direct product of any [[solvability is extension-closed]] || how to construct bigger examples from smaller || if <math>G</math> is a non-nilpotent solvable group with , and <math>H</math> is ''any '' solvable group (possibly nilpotent or , possibly notnilpotent) , <math>G \times H</math> is also a non-nilpotent solvable group . More generally, any extension with normal subgroup <math>G</math> and quotient group <math>H</math>, or with normal subgroup <math>H</math> and get more examples of quotient <math>G</math>, is a non-nilpotent solvable groupsgroup.|}
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