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 \(Sg^{-1}x y\) 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 \(|S| \geq \frac{p-3}{2}\), then \(ind(S) \leq 2\).