关于Erd \H os、Kleitman和Lemke的一个反问题
On an inverse problem of Erd\H os, Kleitman, and Lemke
摘要：
设$(G, 1_G)$是有限群，$S=g_1\bdot \ldots\bdot g_{\ell}$是$G$上的非空序列.我们说$S$是一个微小的积1序列，如果它的各项可以排序，使得它们的乘积等于$1_G$和$\sum_{i=1}^{\ell}\frac{1}{\ord(g_i)}\le 1$。令$\mathsf {ti}(G)$是最小整数$t$，使得每个序列$S$ over $G$ with $|S|\ge t$都有一个极小的积1子序列。正问题是求出$\mathsf {ti}(G)$的精确值，反问题是刻画$G$上不存在微小积1序列的长序列的结构.本文研究了循环群的反问题，同时也研究了二面体群和双循环群的正问题和反问题。
英文摘要：
Let $(G, 1_G)$ be a finite group and let $S=g_1\bdot \ldots\bdot g_{\ell}$ be a nonempty sequence over $G$. We say $S$ is a tiny product-one sequence if its terms can be ordered such that their product equals $1_G$ and $\sum_{i=1}^{\ell}\frac{1}{\ord(g_i)}\le 1$. Let $\mathsf {ti}(G)$ be the smallest integer $t$ such that every sequence $S$ over $G$ with $|S|\ge t$ has a tiny product-one subsequence. The direct problem is to obtain the exact value of $\mathsf {ti}(G)$, while the inverse problem is to characterize the structure of long sequences over $G$ which have no tiny product-one subsequences. In this paper, we consider the inverse problem for cyclic groups and we also study both direct and inverse problems for dihedral groups and dicyclic groups.
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。
