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Math Help - Showing G is Abelian

  1. #1
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    Showing G is Abelian

    Let p & q be primes with q<p and suppose G is a group of order  p^2q . Suppose that G has unique subgroup of order q. Prove that G is abelian stating any theorems you use...

    I'm thinking I have to use some form of Sylows theorem to show that G is a direct sum of p-groups and hence Abelian but no idea how (or even if that is the tactic)!!

    Any help appreciated...
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  2. #2
    Member ModusPonens's Avatar
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    Re: Showing G is Abelian

    Work with the class equation. It involves some work, but it's not dificult.
    Last edited by ModusPonens; April 19th 2012 at 05:43 PM.
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  3. #3
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    Re: Showing G is Abelian

    Oh yeah, before that you need to apply the 3rd Sylow theorem to show that there's only 1 p-Sylow subgroup.
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  4. #4
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    Re: Showing G is Abelian

    Hmmm, I've had a think and come up with this but it isn't really using the class equation I don't think!

    Using Sylow III I can certainly see that n_p = 1 as n_p|q and n_p=1(modp) and as q<p,  n_p =1

    Also from Sylow II we know that as n_p = 1 then  P \triangleleft  G and P has order p^2 thus Abelian. The intersection or P and Q is {e} so G is the direct product of the two abelian groups hence G is Abelian
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  5. #5
    Member ModusPonens's Avatar
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    Re: Showing G is Abelian

    Yes, that works too, as long as you show that G=<QUP>, which, with what you know, is imediate.
    Thanks from leshields
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