  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
$|\Sigma(S)|=19$.
