On minimal zero-sum sequences of length four over cyclic groups
Volume 146 / 2017
Colloquium Mathematicum 146 (2017), 157-163
MSC: Primary 11B50; Secondary 20K01.
DOI: 10.4064/cm6702-1-2016
Published online: 26 August 2016
Abstract
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 $\operatorname{gcd}(n,30)=1$.
