Minimal zero-sum sequence of length five over finite 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 $S=(x_{1}g)\cdot \dots \cdot (x_{l}g)$ 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 $⟨g⟩=G$. Recently the second and the third authors determined the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime order where $S=g^2(x_{2}g)(x_{3}g)(x_{4}g)$. 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 $S=(x_{1}g)(x_{2}g)(x_{2}g)(x_{3}g)(x_{4}g)$ 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 $S=(mg)(mg)(m\frac{n-1}{2}g)(m\frac{n+3}{2}g)(m(n-3)g)$ where $m$ is a positive integer such that $\gcd (m,n)=1$.

References in corpus (1)

• On the index of length four minimal zero-sum sequences

Cited by in corpus (1)

• On the unsplittable minimal zero-sum sequences over finite cyclic groups of prime order