The small sumsets property for solvable finite groups

Let GG be a group written multiplicatively. We say that GG has the small sumsets property   if for all positive integers r,s≤|G|r,s≤|G|, there exist subsets A,B⊂GA,B⊂G such that |A|=r|A|=r, |B|=s|B|=s and |A⋅B|≤r+s−1|A⋅B|≤r+s−1. If, in addition, it is possible to simultaneously satisfy A⊂BA⊂B whenever r≤sr≤s, we speak of the nested small sumsets property for  GG. We prove that finite solvable groups satisfy this stronger form of the property. In the finite non-solvable case, we prove that subsets A,B⊂GA,B⊂G satisfying |A|=r|A|=r, |B|=s|B|=s and |A⋅B|≤r+s−1|A⋅B|≤r+s−1 also exist, provided either r≤12r≤12 or r+s≥|G|−11r+s≥|G|−11.