The small sumsets property for solvable finite groups

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