arXiv 1303.16822013-03-07math.CO

有限循环群II上长度为4的极小零和序列

Minimal zero-sum sequences of length four over finite cyclic groups II

Yuanlin Li, Jiangtao Peng

备注：: 18 pages, accepted for published in Int. J. Number Theory

摘要：

设$G$是有限循环群. $G$上的每个序列$S$都可以写成$S=(n_1g)\cdot\ldots\cdot(n_lg)$的形式，其中$g\in G$和$n_1, \ldots, n_l\in[1, \ord(g)]$，并且$S$的索引$\ind(S)$被定义为所有可能的$g\in G$上的$(n_1+\cdots+n_l)/\ord(g)$的最小值，使得$\langle g \rangle =G$。关于长度为4的序列的索引的一个公开问题是问具有$\gcd(|G|, 6)=1$的有限循环群$G$上的每个长度为4的极小零和序列的索引是否为1。本文证明了：如果$G=\langle g\rangle$是一个循环群，其阶为两个素数幂和$\gcd(|G|, 6)=1$的乘积，则每个形式为$S=(g)(n_2g)(n_3g)(n_4g)$的极小零和序列$S$的指数为1.特别地，我们的结果证实了当$G$的阶是两个不同素数或一个素数幂的乘积时，上述问题有肯定的答案，推广了第一作者Plyley，Yuan和Zeng最近的一个结果.

英文摘要：

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 $n_1, \ldots, n_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be the minimum of $(n_1+\cdots+n_l)/\ord(g)$ over all possible $g\in G$ such that $\langle g \rangle =G$. An open problem on the index of length four sequences asks whether or not every minimal zero-sum sequence of length 4 over a finite cyclic group $G$ with $\gcd(|G|, 6)=1$ has index 1. In this paper, we show that if $G=\langle g\rangle$ is a cyclic group with order of a product of two prime powers and $\gcd(|G|, 6)=1$, then every minimal zero-sum sequence $S$ of the form $S=(g)(n_2g)(n_3g)(n_4g)$ has index 1. In particular, our result confirms that the above problem has an affirmative answer when the order of $G$ is a product of two different prime numbers or a prime power, extending a recent result by the first author, Plyley, Yuan and Zeng.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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