Normal subgroups closed under operation Prove that if N and H are normal subgroups of a group G then the set NH is closed under the operation of G. Hint: Use conjugation.
If N and H are normal subgroups, that means they are invariant under conjugation by G, and $NH=\{nh:n\in N, h\in H\}.$ If $x,y\in NH$, then $xy=n_1h_1n_2h_2$, where $n_1,n_2\in N,h_1,h_2\in H$. I am not sure how to finish this proof, are normal subgroups closed under the operation of G? Because I could show that $NH$ is normal, I don't know if that would be enough.

You want to show that you can write $n_1h_1n_2h_2=nh$, ie as an element of $NH$. But if $H$ is normal, that means you can say $n_2^{-1}h_1n_2=h_3$ for some $h_3\in H$. Rearranging this last equation a bit and subbing into the original will give you your claim. 
Also, to answer your question, every subgroup is closed under the inherited group operation, and this is necessary for the subgroup to be a group in its own right. 
