Proving Abelian Property of Groups Using the Hypothesis ab=ca

[nataliemarie]

I have a homework problem that states: 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.

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?

 

[Norman.Galois]

Suppose G is Abelian and it doesn't have that property. Show a contradiction that G must not be Abelian.

Can you show a contradiction?

 

[nataliemarie]

So I should try assuming its not abelian and try proving that if ab=ca, then b equals c??

 

[Norman.Galois]

So I should try assuming its not abelian and try proving that if ab=ca, then b equals c??

Suppose G is Abelian and it doesn't have that property.

I already wrote the first line for you.

 

[JSuarez]

Start with $$aba^{-1}=c$$ (c is just some element of the group). Now, if you right-multiply by a, and apply the hypothesis, what do you conclude?