Equivalences between fusion systems of finite groups of Lie type

TL;DR

This paper establishes that for certain pairs of finite groups of Lie type, their p-fusion systems are equivalent, using homotopy theory of classifying spaces, revealing deep structural similarities.

Contribution

It proves the equivalence of p-fusion systems for specific pairs of finite groups of Lie type using homotopy theoretic methods, a result not previously known algebraically.

Findings

p-fusion systems are equivalent for certain Lie type groups

Homotopy theory of classifying spaces is used in the proof

No known purely algebraic proof exists for this result

Abstract

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.