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 $S=(x_{1}g)+\cdots +(x_{l}g)$, where $g\in G$ and $x_1,\dots ,x_l\in [1,ord(g)]$, 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 $⟨g⟩=G$. 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 $S=g^2\cdot (x_{2}g)\cdot (x_{3}g)\cdot (x_{4}g)$. 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 $G=⟨g⟩$ is a cyclic group of prime power order $n=p^{\mu }$ with $p\ge 7$ and $\mu \ge 2$, and $S=(x_{1}g)\cdot (x_{2}g)\cdot (x_{3}g)\cdot (x_{4}g)\cdot (x_{5}g)$ 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 $gcd(m,n)=1$.