Let be a subset of , such that . In this paper, we prove that there exist a subgroup H of and a subgroup P of H with |P|≤|H|/8 such that H contains , and is either empty or a full P-coset. We use this result to obtain an upper bound for the cardinality of the subgroup generated by in terms of . More precisely we show that if and then is equal to τ if 1≤τ<7/4, and is less than 8τ/7 if 7/4≤τ<2. This result is optimal.