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

Abstract

Let $p>155$ 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 $S$ is 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 show that if $S$ is an unsplittable minimal zero-sum sequence of length $|S|=\frac{p-1}{2}$, then $S=g^{\frac{p-11}{2}}(\frac{p+3}{2}g{)}^4(\frac{p-1}{2}g)$ or $g^{\frac{p-7}{2}}(\frac{p+5}{2}g{)}^2(\frac{p-3}{2}g)$. Furthermore, if $S$ is a minimal zero-sum sequence with $|S|\ge \frac{p-1}{2}$, then $\ind(S) \leq 2$.

11 pages