Sum-full sets are not zero-sum-free

TL;DR

This paper proves that any finite, sum-full subset of an abelian group necessarily contains a nonempty subset whose elements sum to zero, establishing a fundamental property of such sets.

Contribution

It demonstrates that sum-full subsets in abelian groups cannot be zero-sum-free, providing a new insight into their structural properties.

Findings

Sum-full sets always contain a nonempty zero-sum subset.

Sum-full property implies the existence of zero-sum subsets in abelian groups.

The result applies to all finite, nonempty subsets of abelian groups.

Abstract

Let $A$ be a finite, nonempty subset of an abelian group. We show that if every element of $A$ is a sum of two other elements, then $A$ has a nonempty zero-sum subset. That is, a (finite, nonempty) sum-full subset of an abelian group is not zero-sum-free.