Isometries between finite groups

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 n Ham), thus generalizing previous results of Carlet (1998) for G = Z2 k and Yildiz and Ödemiş Özger (2012) for G = Zp k with p prime. Next, for any positive integer q we define the q-adic metric dq in Zq n and prove that (Zq n , dq) is isometric to (Z n q , dRT ) for every n, where dRT 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 |G| = q n , |K| = q and let H < G. Using chains, we prove that under certain conditions, (G, dC) ≃ (K n , dRT ) and (G, dC) ≃ (H [G:H] , dBRT ) where dBRT is the block Rosenbloom–Tsfasman metric which generalizes dRT .