On Minkowski product size: The Vosper's property

A subset $S$ of a group $G$ is said to be a Vosper's subset if $|A\cup AS|\ge \min (|G|-1,|A|+|S|),$ for any subset $A$ of $G$ with $|A|\ge 2.$ In the present work, we describe Vosper's subsets. Assuming that $S$ is not a progression and that $|S^{-1} S|, |S S^{-1}| <2 |S|,|G'|-1,$ we show that there exist an element $a\in S,$ and a non-null subgroup $H$ of $G'$ such that either $S^{-1}HS =S^{-1}S \cup a^{-1}Ha$ or $SHS^{-1} =SS^{-1}\cup aHa^{-1},$ where $G'$ is the subgroup generated by $S^{-1}S.$