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Non abelian group question

  1. Mar 30, 2012 #1
    To show that a non-abelian group G, has elements x,y,z such that xy = yz where y≠z,

    Is it enough to simply state for non-abelian groups xy≠yx so if you have xy=yz then it is not possible for x=z due to xy≠yx?

    Or is more detail required?
     
  2. jcsd
  3. Mar 30, 2012 #2

    Hurkyl

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    There does not exist any group satisfying that identity.

    In a non-abelian group, only some pairs [itex](a,b)[/itex] satisfy [itex]ab\neq ba[/itex].
     
  4. Mar 30, 2012 #3
    Thanks for replying Hurkyl

    So it would be a similar argument just clarifying that the for some (not all) x,y in G?

    For some x, y in a non-abelian group G

    xy≠yx

    and if

    xy = yz for some z in G

    then x≠z otherwise

    xy≠yx

    Is not satisfied for elements x and y of the group
     
  5. Mar 31, 2012 #4

    chiro

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    To follow on from Hurkyls post, think about identity and inverse elements.
     
  6. Mar 31, 2012 #5
    Do you mean that it should be shown what the is?

    z= y-1x y ?
     
  7. Mar 31, 2012 #6
    If the group is non-abelian, then there sure must be a pair a,b both of which ain't the identity, such that ab != ba. Take such a pair, and play around with the conjugate you proposed.
     
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