A structure theorem for small sumsets in nonabelian groups

Let G be an arbitrary finite group and let S and T be two subsets such that | S | ≥ 2 , | T | ≥ 2 , and | T S | ≤ | T | + | S | − 1 ≤ | G | − 2 . We show that if | S | ≤ | G | − 4 | G | 1 / 2 then either S is a geometric progression or there exists a non-trivial subgroup H such that either | H S | ≤ | S | + | H | − 1 or | S H | ≤ | S | + | H | − 1 . This extends to the nonabelian case classical results for abelian groups. When we remove the hypothesis | S | ≤ | G | − 4 | G | 1 / 2 we show the existence of counterexamples to the above characterization whose structure is described precisely.