arXivDaily arXiv每日学术速递 周一至周五更新
北美退伍军人节（Veterans Day），12 日无更新
arXiv 1303.1676math.CO
收藏

有限循环群上长度为5的极小零和序列

Minimal zero-sum sequences of length five over finite cyclic groups

Jiangtao Peng, Yuanlin Li

备注：
8 Pages. Accepted for publication in Ars Combinatoria
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

设$G$是有限循环群.每个长度为$l$ over $G$的序列$S$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$g\in G$上的最小值$(n_1+\cdots+n_l)/\ord(g)$，使得$\langle g \rangle =G$。当$G=\langle g\rangle$是素数阶循环群且$S$的形式为$S=g^2(n_2g)(n_3g)(n_4g)$时，本文确定了长度为5的极小零和序列$S$的指数.证明了若$G=\langle g\rangle$是素数阶循环群$p \geq 31$，则除$S=g^2(\frac{p-1}{2}g)(\frac{p+3}{2}g)((p-3)g)$外，所有上述形式的极小零和序列$S$的指数为1.

英文摘要：

Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ 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 $\langle g \rangle =G$. In this paper, we determine the index of any minimal zero-sum sequence $S$ of length 5 when $G=\langle g\rangle$ is a cyclic group of a prime order and $S$ has the form $S=g^2(n_2g)(n_3g)(n_4g)$. It is shown that if $G=\langle g\rangle$ is a cyclic group of prime order $p \geq 31$, then every minimal zero-sum sequence $S$ of the above mentioned form has index 1 except in the case that $S=g^2(\frac{p-1}{2}g)(\frac{p+3}{2}g)((p-3)g)$.

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。

↑