One more question i got this proof that seems trivial but im not sure if there was another way to do it. The question is let G and G' be isomorphic groups and G is abelian, prove G' is abelian.

Can you just say that since G and G' are isomorphic they have the same structure so if G is abelian then G' is abelian?

I know that there is a one to one, onto, homomorphism between them so if G is abelian then phi(ab) should = phi(ba) but how would that apply to G'?