Let G be a group. For subsets A, B C G, define AB = {ab | a A and b B}. That is, AB consists of all group elements in G that can be obtained by choosing elements a A, be B and multiplying them (using G's group operation). 1. Show: if H G is a subgroup of G, then HH = H. 2. Show: if G is a finite group, HCG is nonempty, and HH = H, then H is a subgroup of H. 3. Show: there are an infinite group G and a nonempty subset HCG such that HH = H, but H is not a subgroup of G. (You show this last point by giving an example).