HK = KH iff HK is a subgroup of G

Ok so I'm bored and it's late. Time for some random intellectual stimulation. I get enough of that to be honest and I prefer dumb Family Guy episodes or binge watching netflix, but here it is anyways. The setup is as follows.

G a group, and H and K are subgroups of G.


Proposition.
HK is a subgroup of G if and only if HK = KH.

Proof.
Suppose HK is a subgroup of G. Take x in HK, so x = hk, for some h in H and k in K.
Since HK is a subgroup, x^-1 is in HK. Now x^-1 = (hk)^-1 = k^-1h^-1 and this is in KH since k^-1 is in K and h^-1 is in H because K and H are each closed under inverses. This shows HK is a subset of KH. To show KH is a subset of HK the idea is exactly the same, hence omitted!

Conversely, suppose HK = KH. We need to show HK is a subgroup of G. Well first notice e = e*e is in HK because e is in H and e is in K. Next, take x in HK. Since HK = KH, x is in KH, so x = kh for some k in K and h in H. Then x^-1 = h^-1k^-1 and this is in HK because both H and K are closed under inverses. This shows HK is closed under inverses. Finally, suppose x, y are in HK.
This means x = hk and y = h'k' for some h, h' in H and k, k' in K. Then xy = hkh'k'. Now, kh is in HK but HK = KH, so kh = h''k'' for some h'' in H and k'' in K, hence xy = hkh'k' = hh''k''k' and this is in HK because both H and K are closed under the group operation. This shows HK is closed under the group operation and hence HK is a subgroup.