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Prove (a^(-1))^(-1) = a

  1. Dec 17, 2012 #1

    Zondrina

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

    Trying to run through every problem i can in my book in preparation for my exam. I've solved this one before, but it slipped my mind how to do it :

    http://gyazo.com/9fcf9f3cef522c3d5eb1fa7d4ad04394

    2. Relevant equations

    Working in a group, so group axioms I suppose.

    3. The attempt at a solution

    I forgot where to start this one off, I was thinking :

    e = aa-1
    a-1e = a-1aa-1

    That wont get me anywhere though, any pointers would be appreciated.
     
  2. jcsd
  3. Dec 17, 2012 #2

    micromass

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    Do you know that every element in a group has a unique inverse?? You can use this by showing that [itex]a[/itex] and [itex](a^{-1})^{-1}[/itex] are inverses of the same element. So they must be equal.
     
  4. Dec 17, 2012 #3

    Zondrina

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    Ah, so what you're saying is if a is a group element, then it has an inverse which is also a group element denoted by a-1.

    Since a-1 is also a group element and the inverse of a, then (a-1)-1 is also a group element and is the inverse of a-1.
     
  5. Dec 17, 2012 #4

    micromass

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    Yes. So [itex]a^{-1}[/itex] has two inverses. Those inverses must equal.
     
  6. Dec 17, 2012 #5

    I like Serena

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    Hi Zondrina!

    In a group every element has to have an inverse.
    Now suppose we have b=a-1.
    Then according to the group axioms we have: ab=e
    What happens if you multiply the left and right hand sides with b-1?
     
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