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 |TS|≤|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 |HS|≤|S|+|H|−1 or |SH|≤|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.