  We extend the definition of Bockstein basis $\sigma(G)$ to nilpotent groups
$G$. A metrizable space $X$ is called a {\it Bockstein space} if $\dim_G(X) =
\sup\{\dim_H(X) | H\in \sigma(G)\}$ for all Abelian groups $G$. Bockstein First
Theorem says that all compact spaces are Bockstein spaces.
  Here are the main results of the paper: Let $X$ be a Bockstein space. If $G$
is nilpotent, then $\dim_G(X) \leq 1$ if and only if $\sup\{\dim_H(X) |
H\in\sigma(G)\}\leq 1$.
  $X$ is a Bockstein space if and only if $\dim_{\Z_{(l)}} (X) =
\dim_{\hat{Z}_{(l)}}(X)$ for all subsets $l$ of prime numbers.
