Prove that for a, b in G, commutes with , for all integers k, l. Then use the fact that m and n are relatively prime to deduce that k, l can be chosen so that km+ln=1.
You're right, of course. I was overlooking that.
There is a neat trick for proving that , which I found on another forum. The key to it is the calculation , and similarly . Since m and n are coprime, it follows that .