On minimal zero-sum sequences of length four over cyclic groups

Xiangneng Zeng¹ ; Xiaoxia Qi¹

¹ Sino-French Institute of Nuclear Engineering and Technology Sun Yat-Sen University Guangzhou 510275, P.R. China

Colloquium Mathematicum, Tome 146 (2017) no. 2, pp. 157-163

Résumé

Let $G$ be a cyclic group of order $n$. It has been conjectured that if $\operatorname{gcd}(n,6)=1$, then every minimal zero-sum sequence $S$ of length $4$ over $G$ has index $1$, that is, $S=(n_1g)\cdot (n_2g)\cdot (n_3g)\cdot (n_4g)$ for some generator $g\in G$ and some integers $n_1,n_2,n_3,n_4\in [1,n]$ with $n_1+n_2+n_3+n_4=n$. This conjecture has been confirmed recently for the case when $\langle {\rm supp}(S)\rangle =G$ and $S$ contains at least one element $g$ with $\langle g\rangle \not =G$. We show that if $\operatorname{gcd}(n,30)=1$ and any element of $S$ is a generator of $G$, then this conjecture is true. Together with other known results, this conjecture is thus settled in the affirmative when $\operatorname{gcd}(n,30)=1$.

Détail

DOI : 10.4064/cm6702-1-2016

Keywords: cyclic group order has conjectured operatorname gcd every minimal zero sum sequence length has index nbsp cdot cdot cdot generator integers conjecture has confirmed recently langle supp rangle contains least element langle rangle operatorname gcd element generator conjecture together other known results conjecture settled affirmative operatorname gcd