### Subsums of a Zero-sum Free Subset of an Abelian Group

#### Abstract

Let $G$ be an additive finite abelian group and $S \subset G$ a subset. Let f$(S)$ denote the number of nonzero group elements which can be expressed as a sum of a nonempty subset of $S$. It is proved that if $|S|=6$ and there are no subsets of $S$ with sum zero, then f$(S)\geq 19$. Obviously, this lower bound is best possible, and thus this result gives a positive answer to an open problem proposed by R.B. Eggleton and P. Erdős in 1972. As a consequence, we prove that any zero-sum free sequence $S$ over a cyclic group $G$ of length $|S| \ge {6|G|+28\over19}$ contains some element with multiplicity at least ${6|S|-|G|+1\over17}$.