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 (S,F,L) in terms of representation rings of subgroups of S. 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 C∗(|L|∧p ; Fp) → Fp is Gorenstein in the sense of Dwyer–Greenlees–Iyengar and obtain some consequences about the cohomology ring of |L|∧p.