On the index of minimal zero-sum sequences over finite cyclic groups

Let G be a cyclic group of order n >= 2 and S = g(1) . . . . . g(k) a sequence over G. We say that S is a zero-sum sequence if Sigma(k)(i =1) g(i) = 0 and that S is a minimal zero-sum sequence if S is a zero-sum sequence and S contains no proper zero-sum sequence. The notion of the index of a minimal zero-sum sequence (see Definition 1.1) in G has been recently addressed in the mathematical literature. Let vertical bar(G) be the smallest integer t is an element of N such that every minimal zero-sum sequence S over G with length vertical bar S vertical bar >= t satisfies index(S) = 1. In this paper, we first prove that vertical bar(G) = [n/2] + 2 for n >= 8. Secondly, we obtain a new result about the multiplicity and the order of elements in long zero-sumfree sequences. (c) 2007 Elsevier Inc. All rights reserved.