MINIMAL ZERO-SUM SEQUENCES IN FINITE CYCLIC GROUPS

Abstract

Let $C_n$ be the cyclic group of order $n$, $n\geq 20$, and let $S=\prod_{i=1}^k g_i$ be a minimal zero-sum sequence of elements in $C_n$. We say that $S$ is insplitable if for any $g_i\in S$ and any two elements $x,y\in C_n$ satisfying $x+y=g_i$, $Sg_i^{-1}xy$ is not a minimal zero-sum sequence any more. We define $\mbox{Index}(S)=\min_{(m,n)=1} \{\sum_{i=1}^k|mg_i|\}$, where $|x|$ denotes the least positive inverse image under homomorphism from the additive group of integers $\mathbb{Z}$ onto $C_n$. In this paper we prove that for an insplitable minimal zero-sum sequence $S$, if $\mbox{Index}(S)=2n$, then $|S|\leq \lfloor\frac{n}{2}\rfloor+1$.