Minimal Zero-Sum Sequences of Length Five Over Finite Cyclic Groups

Abstract

Let \(G\) be a finite cyclic group. Every sequence \(S\) of length \(l\) over \(G\) can be written in the form \(S = (n_1g) \cdots (n_lg)\), where \(g \in G\) and \(n_1, \ldots, n_l \in [1, \text{ord}(g)]\), and the \({index}\) \(\text{ind}(S)\) of \(S\) is defined to be the minimum of \((n_1 + \cdots + n_l)/\text{ord}(g)\) over all possible \(g \in G\) such that \(\langle g \rangle = G\). In this paper, we determine the index of any minimal zero-sum sequence \(S\) of length \(5\) when \(G = \langle g \rangle\) is a cyclic group of a prime order and \(S\) has the form \(S = g^2{(n_2g)}(n_3g){(n_4)}\). It is shown that if \(G = \langle g \rangle\) is a cyclic group of prime order \(p \geq 31\), then every minimal zero-sum sequence \(S\) of the above-mentioned form has index \(1\), except in the case that \(S = g^2(\frac{p-1}{2}g)(\frac{p+3}{2}g)((p-3)g)\).