Minimal Zero-Sum Sequences of Length Five Over Cyclic Groups of Prime Power Order

Abstract

Let \(G\) be a finite cyclic group. Every sequence \(S\) of length \(l\) over \(G\) can be written in the form \(S = (x_1g) + \cdots + (x_lg)\), where \(g \in G\) and \(x_1, \ldots, x_l \in [1, ord(g)]\), and the index \(ind(S)\) of \(S\) is defined to be the minimum of \((x_1 + \cdots + x_l)/ord(g)\) over all possible \(g \in G\) such that \(\langle g \rangle = G\). Recently, the second and third authors determined the index of any minimal zero-sum sequence \(S\) of length \(5\) over a cyclic group of a prime order where \(S =g^2 \cdot (x_2g)\cdot (x_3g)\cdot (x_4g)\). In this paper, we determine the index of any minimal zero-sum sequence \(S\) of length \(5\) over a cyclic group of a prime power order. It is shown that if \(G = \langle g \rangle\) is a cyclic group of prime power order \(n = p^{\mu}\) with \(p \geq 7\) and \(\mu \geq 2\), and \(S = (x_1g) \cdot (x_2g) \cdot (x_3g) \cdot (x_4g) \cdot (x_5g)\) with \(x_1 = x_2\) is a minimal zero-sum sequence with \(\gcd(n, x_1, x_2, x_3, x_4, x_5) = 1\), then \(ind(S) = 2\) if and only if \(S = (mg) \cdot (mg) \cdot (m\frac{n-1}{2}g) \cdot (m\frac{n+3}{2}g) \cdot (m(n-3)g)\) where \(m\) is a positive integer such that \(\gcd(m,n) = 1\).