Let H and K be distinct subgroups of G. Then the set $\displaystyle HK = \{hk : h \in H, k \in K\}$ 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 $\displaystyle ghg^{-1} \in H$ for all g in G and h in H. I also know that a group H is a subgroup iff $\displaystyle xy^{-1} \in H$ 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?