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 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 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.

Behaving sequences

“…As easy consequences of Theorem 1.2, we shall deduce the following well-known results. [14,17], [8,Theorem 5.1.8].) Let S be a zero-sum-free sequence over a cyclic group of order n 2 with |S| > n 2 .…”

Long unsplittable zero-sum sequences over a finite cyclic group

“…The precise value of I(G) was determined independently by Savchev and Chen [21], and by Yuan [24]. It was shown that I(G) = n 2 + 2 when n ≥ 8.…”

Minimal zero sum sequences of length four over finite cyclic groups

“…It was first addressed by Lemke-Kleitman (in the conjecture [8, page 344]), used as a key tool by Geroldinger [5, page 736], and then investigated by Gao [3] in a systematical way. Since then it has attracted a lot of attention in recent years (see for example [1,2,4,6,7,[9][10][11][12]). …”

“…It was proved independently by Savchen and Chen, and by the third author (see [10,Proposition 10], [12,Theorem 3.1] and [6,Corollary 5.1.9]) that for a finite cyclic group G of order n, if n ∈ {1, 2, 3, 4, 7}, then l(G) = 1; otherwise, l(G) = n 2 + 2. It was proved in [9] that every minimal zero-sum sequence S over G of length 1, 2, 3 has index ind(S) = 1, and if gcd(|G|, 6) = 1 then there exists a minimal zerosum sequence S of length 4 such that S has ind(S) = 1.…”

“…It was conjectured that if gcd(n, 6) = 1 then every minimal zero-sum sequence S of length 4 over G has ind(S) = 1. It can be proved by using the main result of [12] (or a result of Savchen and Chen in [10]) that there exists a minimal zero-sum sequence S of length l with 4 < l < n 2 + 2 in a cyclic group G of order n such that ind(S) = 1. The only unsolved case is that whether or not every minimal zero-sum sequence of length 4 in a cyclic group G with gcd(|G|, 6) = 1 has index 1.…”