Extremal zero-sum sequences without elements of maximal order

Abstract

Let $\mathsf D(G)$ denote the Davenport constant of a finite abelian group $G$, the maximal length of a minimal zero-sum sequence over $G$. Gao and Geroldinger conjectured that every minimal zero-sum sequence of length $\mathsf D(G)$~\cite[Conjecture~6.1]{GG99} and every zero-sumfree sequence of length $\mathsf D(G)-1$~\cite[Conjecture~4.1]{GG06} contains an element of order $\exp(G)$. We disprove both conjectures by proving a density-one theorem: for every fixed finite abelian group $H$ of exponent $M=2^ap^b$, where $a\ge2$, $b\ge1$, and $p$ is an odd prime, the set of integers $r\ge0$ for which $G_r=C_2^r\oplus H$ admits a minimal zero-sum sequence of length $\mathsf D(G_r)$ with no term of order $M$ has natural density one.