On Minkowski product size: The Vosper's property

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