Isometries between finite groups

PODESTÁ, RICARDO A.; VIDES, MAXIMILIANO G.

DISCRETE MATHEMATICS

ELSEVIER SCIENCE BV

Año: 2020 vol. 343

0012-365X

We prove that if H is a subgroup of index n of any cyclic group G then G can be isometrically embedded in (Hn,dHamn), thus generalizing previous results of Carlet (1998) for G=Z2k and Yildiz and Ödemiş Özger (2012) for G=Zpk with p prime. Next, for any positive integer q we define the q-adic metric dq in Zqn and prove that (Zqn,dq) is isometric to (Zqn,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|=qn, |K|=q and let H