arXiv 1402.02212014-02-02math.NT

素数幂阶有限循环群上长度为5的极小零和序列

Minimal zero-sum sequence of length five over finite cyclic groups of prime power order

Li-meng Xia, Yuanlin Li, Jiangtao Peng

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$S=(x_1g)\cdot\ldots\cdot(x_lg)$的形式，其中$g\in G$和$x_1, \ldots, x_l\in[1, \ord(g)]$，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$(x_1+\cdots+x_l)/\ord(g)$，使得$\langle g \rangle =G$。最近，第二和第三作者确定了素数阶循环群上长度为5的极小零和序列$S$的指数，其中$S=g^2(x_2g)(x_3g)(x_4g)$。本文确定了素数幂阶循环群上任意长度为5的极小零和序列$S$的指数。证明了：若$G=\langle g\rangle$是一个具有$p \geq 7$和$μ\geq 2$的素数幂阶$n=p^μ$的循环群，且具有$x_1=x_2$的$S=(x_1g)(x_2g)(x_2g)(x_3g)(x_4g)$是一个具有$\gcd(n,x_1,x_2,x_3,x_4,x_5)=1$的极小零和序列，则$\ind(S)=2$当且仅当$S=(mg)(mg)(m\frac{n-1}{2}g)(m\frac{n+3}{2}g)(m(n-3)g)$其中$m$是一个正整数，使得$\gcd(m,n)=1$ .

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 $S=(x_1g)\cdot\ldots\cdot(x_lg)$ where $g\in G$ and $x_1, \ldots, x_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be the minimum of $(x_1+\cdots+x_l)/\ord(g)$ over all possible $g\in G$ such that $\langle g \rangle =G$. Recently the second and the third authors determined the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime order where $S=g^2(x_2g)(x_3g)(x_4g)$. In this paper, we determine the index of any minimal zero-sum sequence $S$ of length 5 over a cyclic group of a prime power order. It is shown that if $G=\langle g\rangle$ is a cyclic group of prime power order $n=p^μ$ with $p \geq 7$ and $μ\geq 2$, and $S=(x_1g)(x_2g)(x_2g)(x_3g)(x_4g)$ with $x_1=x_2$ is a minimal zero-sum sequence with $\gcd(n,x_1,x_2,x_3,x_4,x_5)=1$, then $\ind(S)=2$ if and only if $S=(mg)(mg)(m\frac{n-1}{2}g)(m\frac{n+3}{2}g)(m(n-3)g)$ where $m$ is a positive integer such that $\gcd(m,n)=1$.

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。

摘要：

英文摘要：

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