Primitive flag-transitive generalized quadrangles

Abstract

Let $\mathcal{Q}$ be a finite thick generalized quadrangle of order $(s,t)$ with a flag-transitive automorphism group $G$. A well-known conjecture of Kantor from the early 1990s asserts that up to duality, $\mathcal{Q}$ is either of classical type, or it is one of two sporadic examples of order $(3,5)$ or $(15,17)$. Although this fundamental problem in finite geometry remains open, it has been resolved in some special cases. For example, a theorem of Buekenhout and Van Maldeghem from 1994 proves Kantor's conjecture under the stronger assumption that $G$ acts distance-transitively on the set of points of $\mathcal{Q}$. In this paper, we study Kantor's conjecture in the setting where $G$ acts primitively on both the points and lines of $\mathcal{Q}$. Here a theorem of Bamberg et al. implies that $G$ is almost simple with socle $X$ a finite simple group of Lie type, and several special cases where the Lie rank of $X$ is at most $2$ have recently been settled in a sequence of papers. Our main theorem resolves the problem for all almost simple groups of Lie type, thereby completing the proof of Kantor's conjecture for generalized quadrangles with a flag-transitive automorphism group acting primitively on points and lines.