If G is a discrete group, we will describe the effect of the BZ/p-nullification and the BZ/p-cellularization over the classifying space of G. These functors have been widely used in the last years for understanding the p-primary part of a space X through the mapping space $map_*(BZ/p,X)$. Our main results are a characterization, in the finite case, of the BZ/p-nullification of BG by means of a Postnikov fibration, a classification of the finite groups whose classifying space is BZ/p-cellular, and in the infinite case, a new relationship between the usual classifying space of G and its classifying space for proper G-bundles by means of a nullification functor. of a nullification functor.