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Math Help - Abelian Subgroups

  1. #1
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    Abelian Subgroups

    Find a non-abelian group whose proper subsets are all abelian.
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    Re: Abelian Subgroups

    which groups have you tried?
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    Re: Abelian Subgroups

    S_3 , GLn
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    Re: Abelian Subgroups

    and what are your conclusions?

    i presume you mean "subgroups" instead of "subsets", because "abelian set" makes no sense at all....
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    Re: Abelian Subgroups

    Quote Originally Posted by veronicak5678 View Post
    S_3 , GLn
    HINT: Try S_3 again...
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    Re: Abelian Subgroups

    aw, i was rooting for you suggesting GL2(F2)....
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    MHF Contributor Swlabr's Avatar
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    Re: Abelian Subgroups

    Quote Originally Posted by Deveno View Post
    aw, i was rooting for you suggesting GL2(F2)....
    Well, I would have suggested a Tarskii Monster Group, but I thought that it was perhaps overkill...
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    Re: Abelian Subgroups

    um, did you not catch the joke i made?
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    MHF Contributor Drexel28's Avatar
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    Re: Abelian Subgroups

    Just a remark, S_3 is definitely a good choice, but why? It's a first-week-of-group-theory matter that every group of order at most five is abelian, thus if you can find a non-abelian group whose proper subgroups all have size at most five, then you're golden. But, clearly any group of order six has the property that every proper subgroup has order at most five, and so in particular, any non-abelian group of order six will have the property you seek. Ta-da, S_3!
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    Re: Abelian Subgroups

    oh you're so cute when you're clever....
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    MHF Contributor Drexel28's Avatar
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    Re: Abelian Subgroups

    Quote Originally Posted by Deveno View Post
    oh you're so cute when you're clever....
    Oh stop!
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    MHF Contributor Swlabr's Avatar
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    Re: Abelian Subgroups

    Quote Originally Posted by Deveno View Post
    aw, i was rooting for you suggesting GL2(F2)....
    Nope, sorry, still don't get it...
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  13. #13
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    Re: Abelian Subgroups

    GL2(F2) is isomorphic to S3
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    MHF Contributor Swlabr's Avatar
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    Re: Abelian Subgroups

    Quote Originally Posted by Deveno View Post
    GL2(F2) is isomorphic to S3
    Oh. Apparently I cannot count - I thought it had 8 elements! (I quickly counted matrices with zero determinant - too quickly...)
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  15. #15
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    Re: Abelian Subgroups

    that's ok, i once proved \mathbb{N}^{\mathbb{N}} was countable, which if correct, would surely have gained me a fields medal.
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