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Group Inverse-Seems to easy, please check!

  1. Oct 17, 2010 #1
    Group Inverse--Seems to easy, please check!

    Hi all, this just seems too easy, and my teacher is prone to typos... I was thinking perhaps it was one.

    1. The problem statement, all variables and given/known data

    Prove that for a group G there exists a unique element that satisfies the equation x * x = x.


    2. The attempt at a solution

    Multiplying both sides by the inverse, x^{-1}, we have


    x^{-1} * x * x = x^{-1} * x => (x^{-1} * x) * x = x^{-1} * x => e * x = e => x = e

    and we see that for this relation to hold, x must be the identity element, which is unique.


    Thanks in advance!
     
  2. jcsd
  3. Oct 17, 2010 #2
    Re: Group Inverse--Seems to easy, please check!

    If you have already proved in class that the identity element must be unique it looks good to me.
     
  4. Oct 18, 2010 #3
    Re: Group Inverse--Seems too easy, please check!

    Thanks!
     
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