Minimal zero-sum sequences of length four over finite cyclic groups II

Abstract

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_{1}g)\cdot \dots \cdot (n_{l}g)$ where $g\in G$ and $n_1, \ldots, n_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be the minimum of $(n_1+\cdots+n_l)/\ord(g)$ over all possible $g\in G$ such that $⟨g⟩=G$. An open problem on the index of length four sequences asks whether or not every minimal zero-sum sequence of length 4 over a finite cyclic group $G$ with $\gcd (|G|,6)=1$ has index 1. In this paper, we show that if $G=⟨g⟩$ is a cyclic group with order of a product of two prime powers and $\gcd (|G|,6)=1$, then every minimal zero-sum sequence $S$ of the form $S=(g)(n_{2}g)(n_{3}g)(n_{4}g)$ has index 1. In particular, our result confirms that the above problem has an affirmative answer when the order of $G$ is a product of two different prime numbers or a prime power, extending a recent result by the first author, Plyley, Yuan and Zeng.

18 pages, accepted for published in Int. J. Number Theory