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=Cn Cn, 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 Cn Cn is said to have Property B, and it is conjectured that this is true for all rank 2 groups Cn Cn. In this paper, we show that Property B is multiplicative, namely, if G=Cn Cn and G=Cm Cm both satisfy Property B, with m, n≥ 3 odd and mn>9, then Cmn Cmn 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.