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Proving a Group is abelian

  1. Jan 28, 2010 #1
    1. The problem statement, all variables and given/known data
    Let G be a group with the following property: Whenever a,b and c belong to G and ab = ca, then b=c. Prove that G is abelian.


    2. Relevant equations



    3. The attempt at a solution
    I started with the hypothesis ab=ca and solved for b and c using inverses. I found b=(a-1)ca and c=ab(a-1). Because the hypothesis says b=c I set them equal. (a-1)ca=ab(a-1). But I'm having trouble getting anywhere useful after that. Hints or suggestions if I'm on the right track???
     
  2. jcsd
  3. Jan 28, 2010 #2

    Landau

    User Avatar
    Science Advisor

    You got two equations from one, so one is redundant. Just stick with [tex]b=a^{-1}ca[/tex]. Now invoke [tex]b=c[/tex], so [tex]b=a^{-1}ba[/tex]. The conclusion follows.
     
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