  In this note we describe the role of the Schur multiplier in the structure of
the $p$-torsion of discrete groups. More concretely, we show how the knowledge
of $H_2G$ allows to approximate many groups by colimits of copies of finite
$p$-groups. Our examples include interesting families of non-commutative
infinite groups, including Burnside groups, certain solvable examples and the
first Grigorchuk group. We also provide a counterexample for a conjecture of E.
Farjoun.
