  Let p, ell be distinct primes and let q be a power of p. Let G be a connected
compact Lie group. We show that there exists an integer b such that the mod ell
cohomology of the classifying space of a finite Chevalley group G(F_q) is
isomorphic to the mod ell cohomology of the classifying space of the loop group
LG for q=p^{ab}, a>0.
