On the index of length four minimal zero-sum sequences

Caixia Shen ¹   ; Li-meng Xia ¹   ; Yuanlin Li ²

¹ Faculty of Science Jiangsu University Zhenjiang, 212013, Jiangsu Prov., China
² Department of Mathematics Brock University St. Catharines, ON Canada L2S 3A1

Colloquium Mathematicum, Tome 135 (2014) no. 2, pp. 201-209

Résumé

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.

Détail

DOI : 10.4064/cm135-2-4

Keywords: finite cyclic group every sequence written form cdot ldots cdot where ldots ord index ind defined minimum cdots ord possible langle rangle conjecture says every minimal zero sum sequence length finite cyclic group gcd has index conjecture confirmed recently product prime powers however general still paper make progress towards solving general langle rangle finite cyclic group order gcd cdot cdot cdot minimal zero sum sequence dots n gcd gcd ind using method much shorter proof index conjecture product prime powers