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Simply group theory problem

  1. Nov 9, 2012 #1
    1. The problem statement, all variables and given/known data
    Let (G,.) be an non-abelian group. Choose distinct x and y such that xy≠yx.

    Show that if x2≠1 then x2[itex]\notin[/itex]{e,x,y,xy,yx}

    3. The attempt at a solution

    If x2=x would imply x.x.x-1=x.x-1 and x=e which cannot be.
    If x2= xy or x2=yx would imply x=y which also cannot be.

    How does one show that x2≠y?
     
  2. jcsd
  3. Nov 9, 2012 #2

    Dick

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    x doesn't commute with y. Does x commute with x^2?
     
  4. Nov 9, 2012 #3
    Yes I believe it does so x2.x=x.x2 which would imply y.x=x.y if x2=y which is a contradiction.

    Thanks for the help. You gave me just enough help to move forward with the problem but left some of the satisfaction of figuring it out to me which is cool, thanks.
     
  5. Nov 9, 2012 #4
    x^2=y→x^3=xy and x^3=yx
     
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