  In this paper we obtain a description of the Grothendieck group of complex
vector bundles over the classifying space of a p-local finite group in terms of
representation rings of subgroups of its Sylow. We also prove a stable elements
formula for generalized cohomological invariants of p-local finite groups,
which is used to show the existence of unitary embeddings of p-local finite
groups. Finally, we show that the augmentation map for the cochains of the
classifying space of a p-local finite group is Gorenstein in the sense of
Dwyer-Greenlees-Iyengar and obtain some consequences about the cohomology ring
of these classifying spaces.
