On the structure of zero-sum free set with minimum subset sums in abelian groups

Let $G$ be an additive abelian group and $S\subset G$ a subset. Let $\mathrm{\Sigma }(S)$ denote the set of group elements which can be expressed as a sum of a nonempty subset of $S$. We say $S$ is zero-sum free if $0\in ̸\mathrm{\Sigma }(S)$. It was conjectured by R.B.~Eggleton and P.~Erd"{o}s in 1972 and proved by W.~Gao et. al. in 2008 that $\mathrm{\mid }\mathrm{\Sigma }(S)\mathrm{\mid }\ge 19$ provided that $S$ is a zero-sum free subset of an abelian group $G$ with $\mathrm{\mid }S\mathrm{\mid }=6$. In this paper, we determined the structure of zero-sum free set $S$ where $\mathrm{\mid }S\mathrm{\mid }=6$ and $\mathrm{\mid }\mathrm{\Sigma }(S)\mathrm{\mid }=19$.