Centers And Finite Coverings Of Finite Loop Spaces

The homotopy theoretic analogue of a compact Lie group is a p--ompact group, i.e a space X with finite mod--p cohomology and an loop structure given by an equivalence of the form X '\Omega BX. The `classifying space' BX has to be a p-complete space. We are concerned with the notions of centers and finite coverings of connected p--compact groups. In particular , we prove in this category two well known results for compact Lie groups; namely that the center of a connected p-compact group is finite iff the fundamental group is finite and that every connected p-compact group has a finite covering which is a product of a simply connected p-compact group and a torus. The latter statement also translates to connected finite loop spaces