Proving Group Abelianity Using Inverse Property

[nataliemarie]

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?

 

[Landau]

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 $$b=a^{-1}ca$$. Now invoke $$b=c$$, so $$b=a^{-1}ba$$. The conclusion follows.