Let G be a group, \(G=G_ 0\supset G_ 1\supset G_ 2\supset..\). be a normal series such that \([G_ i:G_{i+1}]=p\), \(\cap^{\infty}_{i=1}G_ i=\{1\}\), where p is a prime number. Then G is a metric space with the distance function \(\phi(x,y)= e^{- k}\Leftrightarrow xy^{-1}\in G_ k\), \(xy^{-1}\not\in G_{k+1}\). If \(\tilde G=\prod^{\infty}_{i=1}G/G_ i\) then we can define the structure of a metric space on \(\tilde G\) in the following way. If \(x=(x_ 1,x_ 2,...)\), \(y=(y_ 1,y_ 2,...)\in \tilde G\) then \(\rho (x,y)=e^{-k}\) if and only if \(x_ i=y_ i\) \((i=1,2,...,k)\), \(x_{k+1}\neq y_{k+1}\). Let \(\bar G=\{(x_ 1,x_ 2,...)\in \tilde G|\) \(x_{i+1}\subset x_ i\), \(i=1,2,...\}\), \(x^ g=(x_ 1\cdot g,x_ 2\cdot g,...)\) where \(x=(x_ 1,x_ 2,...)\in \tilde G\), \(g\in G\) then the mapping \(\gamma_ g: x\to x^ g\), \(x\in \bar G\) is an isometry.
The main results are the following. Theorem 1. Let \(\psi (g)=\gamma_ g\). Then \(\psi: G\to I_ z\bar G\) (where \(I_ z\bar G\) is the group of isometries of \(\bar G)\) is a continuous monomorphism. - Theorem 2. Let G and \(\psi\) be as above. Then \(\psi(G)\) is a subgroup of the Sylow p- subgroup of \(I_ z\bar G\) and the isometry \(\psi(g)\) does not change \(x_ 1,...,x_ k\) for any \((x_ 1,x_ 2,...)\in G\) if and only if \(g\in G_ k\), \(g\not\in G_{k+1}\). - It immediately follows from the above resuls that in the case of the ring of p-adic numbers \({\mathbb{Z}}_ p\) the following statement is valid.
Corollary. Let G be a countable group which is approximated by finite p- groups. Then G is isomorphic to a subgroup of the Sylow p-subgroup of \(I_ z{\mathbb{Z}}_ p\). In the case when G has two generators a more exact result is announced.