Lee metrics on finite groups

PODESTÁ, RICARDO A.; DENIS E. VIDELA

ADVANCES IN MATHEMATICS OF COMMUNICATIONS

AMER INST MATHEMATICAL SCIENCES

Lugar: springfield; Año: 2023

1930-5346

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 not to have 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 $G imes Z_2^k$ always has Lee metrics for every $k in N$. Then, weshow 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).