# Proving a Group is abelian

1. Jan 28, 2010

### nataliemarie

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. Jan 28, 2010

### Landau

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