Lee metrics on groups

TL;DR

This paper explores interval and Lee metrics on groups, providing conditions for their existence, especially focusing on cyclic, metacyclic, and certain non-cyclic groups, with comprehensive classifications for groups of small order.

Contribution

It characterizes when groups admit Lee metrics, generalizes Lee metrics beyond cyclic groups, and provides a complete classification for small groups.

Findings

Cyclic groups are the only torsion-free or finite odd-order groups with Lee metrics.

Groups admitting Lee metrics can be extended by direct products with elementary abelian 2-groups.

Certain metacyclic groups, including cyclic, dihedral, and dicyclic, always admit Lee metrics.

Abstract

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).