Let (G,⋅) be a finite group. Let H⊆G be a nonempty subset that is stable under products, i.e., for all h,h′∈H we also have h⋅h′∈H. Prove that H is a subgroup of G. Hint: Use the finiteness of G to prove that H being nonempty and closed under products implies that e is in H and that if h is in H then h−1 is also in H.