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

Abstract

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 ∑ i = 1 k 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 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 l ( 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.