  We prove, for certain pairs G,G of finite groups of Lie type, that the
p-fusion systems for G and G' are equivalent. In other words, there is an
isomorphism between a Sylow p-subgroup of G and one of G' which preserves
p-fusion. This occurs, for example, when G=H(q) and G'=H(q') for a simple Lie
type H, and q and q' are prime powers, both prime to p, which generate the same
closed subgroup of the p-adic units. Our proof uses homotopy theoretic
properties of the p-completed classifying spaces of G and G', and we know of no
purely algebraic proof of this result.
