On the Unsplittable Minimal Zero-Sum Sequences Over Finite Cyclic Groups of Prime Order

Abstract

Let $p>165$ be a prime and let $G$ be a cyclic group of order $p$. Let $S$ be a minimal zero-sum sequence with elements over $G$, i.e., the sum of elements in $S$ is zero, but no proper nontrivial subsequence of $S$ has sum zero. We call $S$ unsplittable, if there do not exist $g\in S$ and $x,y\in G$ such that $g=x+y$ and $S{g}^{-1}xy$ is also a minimal zero-sum sequence. In this paper, we determine the structure of $S$ which is an unsplittable minimal zero-sum sequence of length $\frac{p-1}{2}$ or $\frac{p-3}{2}$. Furthermore, if $S$ is a minimal zero-sum sequence with $|S|\ge \frac{p-3}{2}$, then $ind(S)\le 2$.