关于长度为4的极小零和序列的指数
On the index of length four minimal zero-sum sequences
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的序列的指数的一个猜想是说，在具有$\gcd(|G|, 6)=1$的有限循环群$G$上，每个长度为4的极小零和序列的指数为1。该猜想最近被证实的情况下，$|G|$是一个产品的最多两个素数的权力。然而，总的情况仍然开放。在本文中，我们取得了一些进展，解决一般情况。基于以前的工作对这个问题，我们表明，如果$G=\langle g\rangle$是一个有限的循环群的顺序$|G|=n$，使$\gcd(n,6)=1$和$S=(x_1g)(x_2g)(x_3g)(x_4g)$是一个最小的零和序列在$G$，使$x_1,\cdots,x_4\in[1,n-1]$与$\gcd(n,x_1,x_2,x_3,x_4)=1$，和$\gcd(n,x_i)>1$的一些$i\in[1,4]$，然后$\ind(S)=1$。利用本文提出的一种新的证明方法，对$|G|$是两个素数幂的乘积时的指数猜想给出了一个新的证明。
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 $\langle g \rangle =G$. A conjecture on the index of length four sequences says that every minimal zero-sum sequence of length 4 over a finite cyclic group $G$ with $\gcd(|G|, 6)=1$ has index 1. The conjecture was confirmed recently for the case when $|G|$ is a product of at most two prime powers. However, the general case is still open. In this paper, we make some progress towards solving the general case. Based on earlier work on this problem, we show that if $G=\langle g\rangle$ is a finite cyclic group of order $|G|=n$ such that $\gcd(n,6)=1$ and $S=(x_1g)(x_2g)(x_3g)(x_4g)$ is a minimal zero-sum sequence over $G$ such that $x_1,\cdots,x_4\in[1,n-1]$ with $\gcd(n,x_1,x_2,x_3,x_4)=1$, and $\gcd(n,x_i)>1$ for some $i\in[1,4]$, then $\ind(S)=1$. By using an innovative method developed in this paper, we are able to give a new (and much shorter) proof to the index conjecture for the case when $|G|$ is a product of two prime powers.
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