On a classical zero-sum invariant II: Disproof of a long-standing conjecture

Abstract

For a nontrivial finite abelian group $G$, let $\nu (G)$ be the smallest integer $\ell$ such that every zero-sum free sequence $T$ over $G$ of length at least $\ell$ has the following property: all nonzero elements of $G$ that do not occur as a subsequence sum of $T$ lie in a proper coset of some subgroup of $G$. It is easy to check that $\mathsf{d}(G)-1\le \nu (G)\le \mathsf{d}(G)$, where $\mathsf{d}(G)$ is the small Davenport constant of $G$. A conjecture by Gao from the year 2000 stated that equality should always hold at the lower bound. This conjecture has since been confirmed for many families of groups (including all p-groups and groups of rank at most two). In the current note, we disprove the conjecture.