Show that two groups that are isomorphic to a third group are isomorphic to each other

Question

Prove that if G is isomorphic to K and H is isomorphic to K then G and H are isomorphic to each other.

My strategy thus far has been to show that we can use composition of the respective isomorphisms to map from G to K then K to H. Would it be conclusive to show that this process can be inverted thus proving G and H are isomorphic?

Answer 1

Isomorphism is transitive, since the composition of two isomorphisms is an isomorphism. This is utterly straight forward. In fact, it's not hard to see that isomorphism defines an equivalence relation.