Minimal zero-sum sequences of length four over cyclic group with order $n=p^{\alpha }{q}^{\beta }$

Abstract

Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_{1}g)\cdot ...\cdot (n_{k}g)$ where $g\in G$ and $n_1,\cdots ,n_k\in [1,\mathrm{ord}(g)]$, and the index $\ind S$ of $S$ is defined to be the minimum of $(n_1+\cdots +n_k)/\mathrm{ord}(g)$ over all possible $g\in G$ such that $⟨g⟩=G$. A conjecture says that if $G$ is finite such that $\gcd (|G|,6)=1$, then $\ind(S)=1$ for every minimal zero-sum sequence $S$. In this paper, we prove that the conjecture holds if $|G|$ has two prime factors.

Journal of Number Theory (2013). arXiv admin note: text overlap with arXiv:1303.1682, arXiv:1303.1676, arXiv:1401.7981, arXiv:1401.7979