On the index of length four minimal zero-sum sequences
Volume 135 / 2014
Abstract
Let $G$ be a finite cyclic group. Every sequence $S$ over $G$ can be written in the form $S=(n_1g)\cdot\ldots\cdot(n_lg)$ where $g\in G$ and $n_1, \ldots, n_l\in[1, {\rm ord}(g)]$, and the index ${\rm ind}(S)$ is defined to be the minimum of $(n_1+\cdots+n_l)/{\rm ord}(g)$ over all possible $g\in G$ such that $\langle g \rangle =G$. A conjecture says that every minimal zero-sum sequence of length 4 over a finite cyclic group $G$ with ${\rm gcd}(|G|, 6)=1$ has index 1. This conjecture was confirmed recently for the case when $|G|$ is a product of at most two prime powers. However, the general case is still open. In this paper, we make some progress towards solving the general case. We show that if $G=\langle g\rangle$ is a finite cyclic group of order $|G|= n$ such that ${\rm gcd}(n,6)=1$ and $S=(x_1g)\cdot(x_2g)\cdot(x_3g)\cdot(x_4g)$ is a minimal zero-sum sequence over $G$ such that $x_1,\dots,x_4\in[1,n-1]$ with ${\rm gcd}(n,x_1,x_2,x_3,x_4)=1$, and ${\rm gcd}(n,x_i)>1$ for some $i\in[1,4]$, then ${\rm ind}(S)=1$. By using a new method, we give a much shorter proof to the index conjecture for the case when $|G|$ is a product of two prime powers.
