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Thread: Abelian Group 2.12

  1. #1
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    Abelian Group 2.12

    Let {G;*} be a group such that a^2 = e for every a belonging to G. Prove that G must be a abelian group.
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  2. #2
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    Quote Originally Posted by tigergirl View Post
    Let {G;*} be a group such that a^2 = e for every a belonging to G. Prove that G must be a abelian group.
    $\displaystyle
    a.b = b.a.a^{-1}.b^{-1}.a.b$
    $\displaystyle = b.a.a^{-1}.b^{-1}.(b.a).(b.a).a.b$
    $\displaystyle = b.a.e.e$
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  3. #3
    Super Member Gamma's Avatar
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    $\displaystyle a^2=e$
    $\displaystyle b^2=e$
    $\displaystyle abab=(ab)(ab)=(ab)^2=e$
    Now take this and multiply on the left by a and right by b.

    $\displaystyle a(abab)b=a(e)b$
    $\displaystyle (aa)ba(bb)=ab$
    $\displaystyle ba=ab$
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