A classification of finite antiflag-transitive generalized quadrangles

Abstract:

A generalized quadrangle is a point-line incidence geometry $\mathcal {Q}$ such that: (i) any two points lie on at most one line, and (ii) given a line $\ell$ and a point $P$ not incident with $\ell$, there is a unique point of $\ell$ collinear with $P$. The finite Moufang generalized quadrangles were classified by Fong and Seitz [Invent. Math. 21 (1973), 1–57; Invent. Math. 24 (1974), 191–239], and we study a larger class of generalized quadrangles: the antiflag-transitive quadrangles. An antiflag of a generalized quadrangle is a nonincident point-line pair $(P, \ell )$, and we say that the generalized quadrangle $\mathcal {Q}$ is antiflag-transitive if the group of collineations is transitive on the set of all antiflags. We prove that if a finite thick generalized quadrangle $\mathcal {Q}$ is antiflag-transitive, then $\mathcal {Q}$ is either a classical generalized quadrangle or is the unique generalized quadrangle of order $(3,5)$ or its dual. Our approach uses the theory of locally $s$-arc-transitive graphs developed by Giudici, Li, and Praeger [Trans. Amer. Math. Soc. 356 (2004), 291–317] to characterize antiflag-transitive generalized quadrangles and then the work of Alavi and Burness [J. Algebra 421 (2015), 187–233] on “large” subgroups of simple groups of Lie type to fully classify them.