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Quastion about abelian group

  1. Nov 18, 2012 #1
    Hi guys,

    I have quastion about groups:

    G is abelian group with an identity element "e".
    If xx=e then x=e.

    Is it true or false?

    I was thinking and my feeling is that it's true but I just can't prove it.


    I started with:

    (*) ae=ea=a
    (*) aa^-1 = a^-1 a = e
    those from the definition of Group

    and now the assuming: aa=e

    then:

    aa^-1 = e = aa
    a=a^-1
    ==> a^-1 a = aa = e

    that's all i got.
    Is anyone can halp?

    thank you!
     
  2. jcsd
  3. Nov 18, 2012 #2
    what about the additive group of integers modulu 4?2 has order 2.
     
  4. Nov 18, 2012 #3

    micromass

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    If G is finite, then you can prove that your result is true if and only if G has odd order.
     
  5. Nov 19, 2012 #4
    But the main argument is about ALL abelian group with xx=e
     
  6. Nov 19, 2012 #5

    HallsofIvy

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    To disprove a general statement, you only need one counterexample.
     
  7. Nov 19, 2012 #6
    Do you mean grouo af all integers -
    the identity element is 0
    and for example 2 +(mod4) 2 = 0
    although 2 ≠ 0
    (I still trying to understand the modulo)
     
  8. Nov 19, 2012 #7

    HallsofIvy

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    Yes, that is what he meant.
     
  9. Nov 19, 2012 #8
    Thank you so much! :)
     
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