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

Let G be a cyclic group of order n⩾2 and a sequence over G. We say that S is a zero-sum sequence if 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 l(G) be the smallest integer t∈N such that every minimal zero-sum sequence S over G with length |S|⩾t satisfies index(S)=1. In this paper, we first prove that for n⩾8. Secondly, we obtain a new result about the multiplicity and the order of elements in long zero-sumfree sequences.