On p-completed classifying spaces of discrete groups and finite complexes

It is shown that for certain discrete p-perfect groups G, in particular for all p-perfect finite groups and certain groups of finite virtual cohomological dimension, the loop space on the p-completed classifying space Omega BG(p)(A) is a retract of the loop space on the p-completion of a certain finite complex. For those finite virtual cohomological dimension groups where the author's methods apply, a bound on the dimension of such a complex is provided.