Equivalences between fusion systems of finite groups of Lie type

Abstract

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