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 \(\Sigma(S)\) 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 \(0 \not\in \Sigma(S)\). It was conjectured by R.B.~Eggleton and P.~Erd\"{o}s in 1972 and proved by W.~Gao et. al. in 2008 that \(|\Sigma(S)|\geq 19\) provided that \(S\) is a zero-sum free subset of an abelian group \(G\) with \(|S|=6\). In this paper, we determined the structure of zero-sum free set \(S\) where \(|S|=6\) and \(|\Sigma(S)|=19\).