True of false: For every integer there exists a non-abelian group of order

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- July 2nd 2009, 12:13 PMNonCommAlgAlgebra, Problems For Fun (30)
__True of false__: For every integer there exists a non-abelian group of order - July 2nd 2009, 03:58 PMBruno J.
If is even, is nonabelian of order .

If is odd, perhaps we can decompose into a product of prime powers and show that for every odd prime there exists a nonabelian group of order ... (Worried) - July 3rd 2009, 12:17 AMSwlabr
It is sufficient to prove the result for , a prime number, as then cross products give us the result.

So, a non-abelian group of order ? Does the group with presentation work? Clearly it is non-abelian (as if it was ), and it has order (as with a minimal generating set and ).

(This is a specific case of the group , , the only non-abelian p-group that has a cyclic maximal subgroup and is not of maximal class - see Robinson, A Course in the Theory of Groups, section 5.3.4) - July 3rd 2009, 06:27 PMNonCommAlg
- July 4th 2009, 12:25 AMSwlabr