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- W4287975099 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 $d_{RT}$." @default.
- W4287975099 created "2022-07-26" @default.
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- W4287975099 date "2020-01-01" @default.
- W4287975099 modified "2023-09-27" @default.
- W4287975099 title "Isometries between finite groups" @default.
- W4287975099 doi "https://doi.org/10.48550/arxiv.2001.00213" @default.
- W4287975099 hasPublicationYear "2020" @default.
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