For 1, you want to prove that K is isomorphic to a subgroup of H. So you need to exhibit an isomorphism between them. The only map you've got to work with is Theta. Consider restricting Theta to a map from K to H. This is an isomorphiosm between K and the image im(Theta) if and only if it is one-to-one (injective). A homomorphism is injective if and only if it has kernel equal to {1}. What can you say about the kernel of the restriction of Theta to K?