Equivalences between fusion systems of finite groups of Lie type

Abstract

We prove, for certain pairsG,G′G,G’of finite groups of Lie type, that thepp-fusion systemsFp(G)\mathcal {F}_p(G)andFp(G′)\mathcal {F}_p(G’)are equivalent. In other words, there is an isomorphism between a Sylowpp-subgroup ofGGand one ofG′G’which preservespp-fusion. This occurs, for example, whenG=G(q)G=\mathbb {G}(q)andG′=G(q′)G’=\mathbb {G}(q’)for a simple Lie “type”G\mathbb {G}, andqqandq′q’are prime powers, both prime topp, which generate the same closed subgroup ofpp-adic units. Our proof uses homotopy-theoretic properties of thepp-completed classifying spaces ofGGandG′G’, and we know of no purely algebraic proof of this result.