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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 augmentationC  ∗(  |  L  |   p  ∧   ;   F  p    )  →    F  pis Gorenstein in the sense of Dwyer–Greenlees–Iyengar and obtain some consequences about the cohomology ring of|  L  |   p  ∧   .

Vector bundles over classifying spaces of p ‐local finite groups and Benson–Carlson duality