Inverse Zero-Sum Problems III

Abstract

Let $G$ be a finite abeilian group. A sequence $S$ with terms from $G$ is zero-sum if the sum of terms in $S$ equals zero. It is a minimal zero-sum sequence if no proper, nontrivial subsequence is zero-sum. The maximal length of a minimal zero-sum subsequence in $G$ is the Davenport constant, denoted $D(G)$. For a rank 2 group $G=C_n\oplus C_n$, it is known that $D(G)=2n-1$. However, the structure of all maximal length minimal zero-sum sequences remains open. If every such sequence contains a term with multiplicity $n-1$, then $C_n\oplus C_n$ is said to have Property B, and it is conjectured that this is true for all rank 2 groups $C_n\oplus C_n$. In this paper, we show that Property B is multiplicative, namely, if $G=C_n\oplus C_n$ and $G=C_m\oplus C_m$ both satisfy Property B, with $m,n\ge 3$ odd and $mn>9$, then $C_{mn}\oplus C_{mn}$ satisfies Property B also. Combined with previous work in the literature, this reduces the question of establishing Property B to the prime cases, and in such case the complete structural description of the sequence follows.