Isometries between finite groups

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 = Z(2k) and Yildiz and odemis ozger (2012) for G = Z(pk) with p prime. Next, for any positive integer q we define the q-adic metric d(q) in Z(qn) and prove that (Z(qn), d(q)) is isometric to (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 C of subgroups of a given group and define the chain metric dC and chain isometries between two chains. Let G, K be groups with vertical bar G vertical bar = qn, vertical bar K vertical bar = q and let H < G. Using chains, we prove that under certain conditions, (G, d(C)) similar or equal to (K-n, d(RT)) and (G, d(C)) similar or equal to (H-[G:H], d(BRT)) where d(BRT) is the block Rosenbloom-Tsfasman metric which generalizes d(RT). (C) 2020 Elsevier B.V. All rights reserved.