论文标题
Lee指标在小组上
Lee metrics on groups
论文作者
论文摘要
在这项工作中，我们考虑对小组的间隔指标；也就是说，相关权重函数没有间隙的整体不变指标。我们为一个小组拥有并且没有间隔指标提供条件。然后，我们研究了一般组的Lee指标，即具有最好的单一对称相关分区的间隔指标。这些指标将经典的Lee度量推广到循环基团上。如果$ g $是无扭力的组或有限的奇数组，我们证明$ g $在且仅当$ g $是循环时才具有Lee度量。另外，如果$ g $是接纳Lee指标的集团，那么$ g \ times \ mathbb {z} _2^k $总是每$ k \ in \ mathbb {n} $都有lee指标。然后，我们表明，一些元环体群体（例如循环，二面和双环群）总是具有Lee指标。最后，我们提供了非循环群体的条件，使它们没有Lee指标。我们以所有订单$ \ le 31 $组的表结尾，指示其中哪些具有（或没有）Lee指标以及为什么（不）。
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).
