  The theory of p-local compact groups, developed in an earlier paper by the
same authors, is designed to give a unified framework in which to study the
p-local homotopy theory of classifying spaces of compact Lie groups and
p-compact groups, as well as some other families of a similar nature. It also
includes, and in many aspects generalizes, the earlier theory of p-local finite
groups. In this paper we show that the theory extends to include classifying
spaces of finite loop spaces. Our main theorem is in fact more general and
states that in a fibration whose base spaces if the classifying space of a
finite group, and whose fibre is the classifying space of a p-local compact
group, the total space is, up to p-completion the classifying space of a
p-local compact group.
