Lee群度量
Lee metrics on groups
摘要：
在这项工作中，我们认为区间度量组，即积分不变度量的相关权函数没有差距。我们给出了一个组有和没有区间度量的条件。然后我们研究了一般群上的Lee度量，即具有最优酉对称伴随分拆的区间度量。这些度量推广了循环群上经典的Lee度量.在$G$是无挠群或奇数阶有限群的情况下，证明了$G$有Lee度量当且仅当$G$是循环的.而且，如果$G$是准许Lee度量的组，则$G \times \mathbb{Z}_2^k$总是具有用于每个$k \in \mathbb{N}$的Lee度量。然后，我们证明了某些亚循环群族，如循环群、二面体群和双循环群，总是具有Lee度量.最后，给出了非循环群不具有Lee度量的条件.最后，我们用$\le 31$阶的所有组的表来表示它们中的哪些具有（或不具有）Lee度量以及为什么（不具有）。
英文摘要：
In this work we consider interval metrics on groups; that is, integral invariant metrics whose associated weight functions do not have gaps. We give conditions for a group to have and to have not interval metrics. Then we study Lee metrics on general groups, that is interval metrics having the finest unitary symmetric associated partition. These metrics generalize the classic Lee metric on cyclic groups. In the case that $G$ is a torsion-free group or a finite group of odd order, we prove that $G$ has a Lee metric if and only if $G$ is cyclic. Also, if $G$ is a group admitting Lee metrics then $G \times \mathbb{Z}_2^k$ always have Lee metrics for every $k \in \mathbb{N}$. Then, we show that some families of metacyclic groups, such as cyclic, dihedral, and dicyclic groups, always have Lee metrics. Finally, we give conditions for non-cyclic groups such that they do not have Lee metrics. We end with tables of all groups of order $\le 31$ indicating which of them have (or have not) Lee metrics and why (not).
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。
