Hello!

If we have a group G: if $\displaystyle a,b,c \in G$

and $\displaystyle ab=ca$ implies $\displaystyle b=c$

then prove G is Albelian.

To me, it seems trivial, that if we assume ab = ca and since b=c, $\displaystyle ab = ba $ $\displaystyle \forall a,b \in G$

I have a feeling this proof is not valid, but I am not sure how? Does anyone else see an error in this logic, or have a suggestion for a method to prove this properly? Or is this legit?

Thank you!