A classification of finite antiflag-transitive generalized quadrangles

TL;DR

This paper classifies finite antiflag-transitive generalized quadrangles, showing they are either classical or of a specific small order, expanding understanding beyond previously classified Moufang cases.

Contribution

It extends classification to antiflag-transitive generalized quadrangles, identifying all such structures as either classical or of order (3,5) or its dual.

Findings

Finite antiflag-transitive generalized quadrangles are either classical or of order (3,5) or its dual.

The classification broadens understanding beyond Moufang quadrangles.

The result confirms the uniqueness of certain small-order examples.

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 (1973), and we study a larger class of generalized quadrangles: the \emph{antiflag-transitive} quadrangles. An antiflag of a generalized quadrangle is a non-incident 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.