C. Broto, J. M. M\o{}ller, and B. Oliver

Equivalences between fusion systems of finite groups of Lie type

We prove, for certain pairs $G,G'$ of finite groups of Lie type, that the 
$p$-fusion systems $F_p(G)$ and $F_p(G')$ 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 
$G=\Gamma(q)$ and $G'=\Gamma(q')$ for a simple Lie ``type'' $\Gamma$, and 
$q$ and $q'$ are prime powers, both prime to $p$, which generate the same 
closed subgroup of $p$-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.