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Math Help - show that G is abelian....

  1. #1
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    Lightbulb show that G is abelian....

    If G is a group such that (ab)= a2b2 forevery a,b Є G, show that G is abelian.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: show that G is abelian....

    Quote Originally Posted by Aman3230 View Post
    If G is a group such that (ab)= a2b2 forevery a,b Є G, show that G is abelian.
    Surely you meant $(ab)^2=a^2b^2.$ Using that the group product is cancellable:

    $$(ab)^2=a^2b^2\Rightarrow abab=aabb\Rightarrow bab=abb\Rightarrow ab=ba\;(\forall a,b\in G.)$$
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  3. #3
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    Re: show that G is abelian....

    thanks 4 reply.....




    in my text book only (ab)= a2b2 was given anyway

    thanks alot 4 giving ur precious time
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  4. #4
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    Re: show that G is abelian....

    If ab = a2b2 for every a,b in G, then it certainly is the case for a = e, the identity of G.

    Then we have:

    eb = e2b2

    b = b2

    e = b, for EVERY b in G, which means that G is a group with only one element: e.
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  5. #5
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    Re: show that G is abelian....

    thanks
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