  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 $d_{RT}$.
