Invariant metrics on finite groups

TL;DR

This paper classifies and counts invariant and bi-invariant metrics on finite groups, establishing a correspondence with symmetric partitions and providing explicit counts for various group types.

Contribution

It introduces a novel classification of invariant metrics via symmetric partitions and derives formulas for counting these metrics on finite groups.

Findings

Non-equivalent invariant metrics correspond to symmetric partitions on groups.

Number of metrics expressed using Bell numbers for certain groups.

Characterization of groups where all invariant metrics are bi-invariant.

Abstract

We study invariant and bi-invariant metrics on groups focusing on finite groups $G$. We show that non-equivalent (bi) invariant metrics on $G$ are in 1-1 correspondence with unitary symmetric (conjugate) partitions on $G$. To every metric group $(G,d)$ we associate to it the symmetry group and the weighted graph of distances. Using these objects we can classify all equivalence classes of invariant and bi-invariant metrics for small groups. We then study the number of non-equivalent invariant and bi-invariant metrics on $G$. We give an expression for the number of such metrics in terms of Bell numbers, with closed expressions for certain groups such as abelian, dihedral, quasidihedral and dicyclic groups. We then characterize all the groups (finite or not) in which every invariant metric is also bi-invariant. We give the number of non-equivalent invariant and bi-invariant metrics for all the groups of order up to 32.