Isometries between finite groups

TL;DR

This paper explores isometric embeddings of finite groups into metric spaces, generalizing previous results and introducing new metrics like the chain metric, with applications to group structures and coding theory.

Contribution

It introduces new isometric embeddings of finite groups into metric spaces, generalizes existing results, and constructs explicit metrics like the chain metric for group analysis.

Findings

Finite groups of the same size can be isometric under explicitly constructed metrics.

Any pair of finite groups of equal order are isometric to each other with some metric.

Chain metrics can relate different group structures through isometries.

Abstract

We prove that if $H$ is a subgroup of index $n$ of any cyclic group $G$, then $G$ can be isometrically embedded in $(H^n, d_{_{Ham}}^n)$, thus generalizing previous results of Carlet (1998) for $G=\mathbb{Z}_{2^k}$ and Yildiz-\"Ozger (2012) for $G=\mathbb{Z}_{p^k}$ with $p$ prime. Next, for any positive integer $q$ we define the $q$-adic metric $d_q$ in $\mathbb{Z}_{q^n}$ and prove that $(\mathbb{Z}_{q^n}, d_q)$ is isometric to $(\mathbb{Z}_q^n, d_{RT})$ for every $n$, where $d_{RT}$ is the Rosenbloom-Tsfasman metric. More generally, we then demonstrate that any pair of finite groups of the same cardinality are isometric to each other for some metrics that can be explicitly constructed. Finally, we consider a chain $\mathcal{C}$ of subgroups of a given group and define the chain metric $d_{\mathcal{C}}$ and chain isometries between two chains. Let $G, K$ be groups with $|G|=q^n$, $|K|=q$ and let $H<G$. Using chains, we prove that under certain conditions, $(G,d_\mathcal{C}) \simeq (K^n, d_{RT})$ and $(G,d_\mathcal{C}) \simeq (H^{[G:H]}, d_{BRT})$ where $d_{BRT}$ is the block Rosenbloom-Tsfasman metric which generalizes $d_{RT}$.