On Subset Sums Of A Zero-Sum Free Set Of Seven Elements From An Abelian Group

Let G be a finite abelian group and S subset of 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 is not an element of Sigma(S). Suppose S is zero-sum free with vertical bar S vertical bar= 7. It was proved by P. Yuan and X. Zeng in 2010 that vertical bar Sigma(S)vertical bar >= 24. We show that if < S > is not cyclic, then vertical bar Sigma(S)vertical bar >= 25. Furthermore if vertical bar Sigma(S)vertical bar = 24 then < S > is a cyclic group and 25 vertical bar vertical bar G vertical bar, which supports a conjecture of R. B. Eggleton and P. Erdos.