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Group theory question

  1. Oct 4, 2011 #1
    1. The problem statement, all variables and given/known data

    If a and b are in a group, show that if (ab)^n=e then (ba)^n=e.

    2. Relevant equations



    3. The attempt at a solution

    I'm not sure how one would prove this. The question is obviously for non-abelian groups.
     
  2. jcsd
  3. Oct 4, 2011 #2

    Dick

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    If (ab)^2=e then (ab)(ab)=e. So b(ab)(ab)a=bea=ba. Now use associativity and 'cancellation'. Do you see how to do the same trick for (ab)^n?
     
  4. Oct 4, 2011 #3
    I'm not sure how this helps us show that (ba)^2=e? When generalizing to (ab)^n I see we'll get a similar result but I'm not sure how this shows that (ba)^n=e.
     
  5. Oct 4, 2011 #4

    Dick

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    b(ab)(ab)a=ba, yes? That's the same as (ba)(ba)(ba)=(ba). Do you see it now?
     
  6. Oct 4, 2011 #5
    abab=e , so ab=b-1a-1

    ababab=e , so baba=a-1b-1

    Play around with these until you can figure out one, then , if you don't have a general
    argument for all n, maybe induction on n will help.
     
  7. Oct 4, 2011 #6
    Oh, ok. Now I understand the argument. Thank you.
     
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