Proving a Group is abelian

  • #1

Homework Statement


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.


Homework Equations





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???
 

Answers and Replies

  • #2
Landau
Science Advisor
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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).
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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