I think I'm on the brink of seeing the solution. Are you saying that if HxK is commutative then G must also be commutative in order for HxK to be isomorphic to G? And if so, why is this?
Two groups are isomorphic if they are the same `up to labelling'. They are essentially the same, just every element has a different name. For example, you can think of the integers as under addition, or you can think of them as under multiplication. They are still the same group!
Basically, if and are isomorphic then let be the isomorphism. As for all we must have that in . Aka, . As is onto, you are done.