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

Abstract

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 $0\notin \Sigma (S)$. It was conjectured by R.B.~Eggleton and P.~Erdös in 1972 and proved by W.~Gao et. al. in 2008 that $|\Sigma (S)|\ge 19$ 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 $|\Sigma (S)|=19$.

19 pages