Sequences not containing long zero-sum subsequences

Abstract

Let G be a finite abelian group (written additively), and let D ( G ) denote the Davenport’s constant of G , i.e. the smallest integer d such that every sequence of d elements (repetition allowed) in G contains a nonempty zero-sum subsequence. Let S be a sequence of elements in G with | S | ≥ D ( G ) . We say S is a normal sequence if S contains no zero-sum subsequence of length larger than | S | − D ( G ) + 1 . In this paper we obtain some results on the structure of normal sequences for arbitrary G . If G = C n ⊕ C n and n satisfies some well-investigated property, we determine all normal sequences. Applying these results, we obtain correspondingly some results on the structure of the sequence S in G of length | S | = | G | + D ( G ) − 2 and S contains no zero-sum subsequence of length | G | .