Bockstein Theorem for Nilpotent Groups

We extend the definition of Bockstein basis sigma(G) to nilpotent groups G. A metrizable space X 
is called a 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:

Theorem: Let X be a Bockstein space. If G is nilpotent, then dim_G(X) <= 1 if and only if sup{dim_H(X) | H in sigma(G)} <= 1.

Theprem: 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.