Abstract Let p be a fixed prime number. Let G denote a finite p-perfect group. We study the homotopy type of the p-completed classifying space BGp. The pa- per is divided into two parts. In part 1 we study the homology, homotopy and stable homotopy of QBG^, where G is a finite p-perfect group. We construct an algebraic analogue of the Quillen's "plus" construction for differential gra- ded coalgebras. This construction is used to show that given a finite p-perfect group G, the loop spaces ftBG^ admits integral homology exponents. We give examples to show that in some cases our bound is best possible. We show that in general Bir£ admits infinitely many non-trivial /c-invariants and discuss some examples where homotopy exponents exist. Finally classical constructions in stable homotopy theory are used to show that the stable homotopy groups of these loop spaces also have exponents. In Part 2 we define the concept of resolutions by fibrations over an arbitrary family of spaces. We construct examples for finite p-perfect groups G such that QBGp is finitely resolvable over the family of spheres and their iterated loop spaces. In particular we get such resolutions for loop spaces associated with most finite groups of Lie type and with some of the polynomial cohomology spaces of Ewing and Clark. Various sporadic examples are also discussed. Key words and phrases. Finite Groups, Completion and Localization, Loop Spaces, Ex- ponents, Spherical resolvability. The author was supported in part by NSF Grant #DMS-9013139.

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