On subset sums of a zero-sum free set ofseven elements from an abelian group
Let $G$ be a finite abelian group and $S\subset G$. Let $\Sigma (S)$ denote the set of group elements which can be expressed as a sum of a nonempty subset of $S$. We say that $S$ is zero-sum free if $0\notin \Sigma (S)$. Suppose $S$ is zero-sum free with $|S|=7$. It was proved by P. Yuan and X. Zeng in 2010 that $|\Sigma (S)|\geq 24$. We show that if $\langle S\rangle $ is not cyclic, then $|\Sigma (S)|\geq 25$. Furthermore if $|\Sigma (S)|= 24$ then $\langle S\rangle $ is a cyclic group and $25 \,|\, |G|$, which supports a conjecture of R. B. Eggleton and P. Erdős.