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Abstract - one sided identity question

  1. Mar 2, 2009 #1
    1. The problem statement, all variables and given/known data

    I'm trying to find what a a left and right identity element is.
    Also, I want to see if a one sided element for * exists, if it is unique.

    2. Relevant equations



    3. The attempt at a solution
    Ok, I just don't really know what a one sided element is.
    I'm using e*s=s*e=s, but is e*s the left identity?
    I think my problem with finding uniqueness is getting started and the fact that I don't really understand what a one sided identity element is?
     
  2. jcsd
  3. Mar 2, 2009 #2

    CompuChip

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    e is a left identity, if e * s = s for all s.
    It is a right identity if s * e = s for all s.
    Usually, when speaking about "an identity element" we mean that it's both left and right-handed (e * s = s * e = e), although it can be shown that it suffices to demand only a left-handed identity (for example) in the group axioms, which will then automatically also be a right-handed identity element.

    For the uniqueness: suppose you have two of them, e and e'.
    Then e * s = s = e' * s.
     
  4. Mar 2, 2009 #3
    ok, so for uniqueness, I use e*s=e'*s and show that e=e' for uniqueness?
     
  5. Mar 2, 2009 #4

    CompuChip

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    Yep.
    That's actually a very standard way of proving uniqueness (assume that there are two things with the defining property and show that they must be the same).
     
  6. Mar 2, 2009 #5
    Thanks, that makes sense!
     
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