"Then the set is a subgroup of G by the closure of a group."
What do you mean by this? Isn't this the very thing you are trying to prove?
Let H and K be distinct subgroups of G. Then the set is a subset of G by the closure of a group. Prove that HK is a subgroup of G if and only if at least one of H or K is a normal subgroup of G. I'm just kind of stuck with this - I know that a subgroup H is a normal subgroup of G iff for all g in G and h in H. I also know that a group H is a subgroup iff for all x,y in H. But I'm having trouble using the definition of HK to show these properties, both in the forward and backward direction of the "if and only if". Can someone just point me in the right direction?
Ok, here's my proof for the right-to-left direction of the statement ("if at least one of H or K is a normal subgroup of G, then HK is a subgroup of G"):
Assume H is a normal subgroup of G, and let and , so that . Then . But since H is a normal subgroup, then , so by closure of a group, and by closure of a group, so , and HK is closed under the group operation. An analogous argument can be made if K is assumed to be a normal subgroup of G. Now let and so that . Then , and since K is a subgroup of G, then , and since H is a subgroup, , so and HK is closed under inverses. Since HK is closed under the group operation and inverses, HK is a subgroup of G.
I think that's pretty sound. I'm not sure how to go in the other direction though: If HK is a subgroup of G, then at least one of H or K is a normal subgroup of G.