  One way to understand the mod p homotopy theory of classifying spaces of
finite groups is to compute their BZ/p-cellularization. In the easiest cases
this is a classifying space of a finite group (always a finite p-group). If
not, we show that it has infinitely many non-trivial homotopy groups. Moreover
they are either p-torsion free or else infinitely many of them contain
p-torsion. By means of techniques related to fusion systems we exhibit concrete
examples where p-torsion appears.
