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Proof of the inverse of an inverse

  1. Sep 13, 2008 #1
    1. The problem statement, all variables and given/known data
    In any group, verify directly from the axioms that
    (a) inverse of the inverse of x= x
    (b) (xy)^inverse = (inverse y)(inverse x) for all x,y in G. (note the reversal here)


    3. The attempt at a solution
    (a) I tried to use the axiom that xe=x=ex but I don't know where to go from there.
    (b) I don't know how to start it.
     
  2. jcsd
  3. Sep 13, 2008 #2

    Dick

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    If b is the inverse of a then ab=ba=e. If a is the inverse of b then ba=ab=e. They are the SAME THING. Think of what that means if a=x and b=x^(-1).
     
  4. Sep 13, 2008 #3
    So my proof should conclude with noticing that x is the inverse of x-inverse?
     
  5. Sep 13, 2008 #4

    Dick

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    Well, yes. It is, isn't it?
     
  6. Sep 13, 2008 #5

    statdad

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    For the second question - what happens if you multiply [tex] xy [/tex] with the object you need to show is its inverse?
     
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