I am new to algebra and I'm doing some proofs for fun. I came across this one. Does anyone care to check my proof? It seems fine to me but I am thrown off by the fact that I didn't use that G is abelian. The hint also throws me off. I understand that if I proceeded that way, I would need to use that G is abelian to prove that phi is a homomorphism, but it doesn't seem necessary to me.