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Group Theory (Abstract algebra manipulations)

  1. Aug 4, 2011 #1
    1. The problem statement, all variables and given/known data
    Let a,b,c,d be elements of a group G and let ab = c, bc = d, cd = a, da = b. Examine the expression da^2b and first derive an expression for b in powers of a. Then express c and d in powers of a. Show that a^5 = e (identity element)


    2. Relevant equations



    3. The attempt at a solution
    So ive been banging me head against this one for a while now. I feel like its something simple and obvious but i cant figure it out. I have an expression for b:

    da^2b = daab = (da)(ab) = bc = d

    d^-1*daab=d^-1*d
    eaab = e
    b = a^-2

    I cant figure out c or d, well i should say I have worked out values for c and d but know them to be wrong because I can't show a^5 = e. (and also I have the answers to this question)

    Thanks
     
  2. jcsd
  3. Aug 4, 2011 #2

    micromass

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    You know c=ab. So what happens if you substitute b=a-2 in that equation?
     
  4. Aug 4, 2011 #3
    Well c = a*a^-2
    c=a^-1 right?

    Thats the value I got but when I look at the solution its wrong.

    Supposedly c = b^-2 = a^4
     
  5. Aug 4, 2011 #4

    micromass

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    Indeed, that is what we want to prove. You already know that [itex]c=a^{-1}[/itex]. And what we want to prove is that [itex]a^{-1}=a^{4}[/itex]. But to prove that, you will want to calculate d first...
     
  6. Aug 4, 2011 #5
    Oh, I just worked it out. God I feel dumb.

    Thx for your help micromass :)
     
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