# A classification of finite locally 2-transitive generalized quadrangles

## TL;DR

This paper advances the classification of finite generalized quadrangles by analyzing automorphism groups with specific transitivity properties, contributing to the understanding of their structure and supporting longstanding conjectures.

## Contribution

It extends previous classifications by considering automorphism groups transitive on pairs of collinear points and concurrent lines, moving closer to resolving the Kantor conjecture.

## Key findings

- Classified generalized quadrangles with automorphism groups transitive on pairs of collinear points.
- Identified new structural properties of these quadrangles.
- Provided evidence supporting the uniqueness of classical examples.

## Abstract

Ostrom and Wagner (1959) proved that if the automorphism group $G$ of a finite projective plane $\pi$ acts $2$-transitively on the points of $\pi$, then $\pi$ is isomorphic to the Desarguesian projective plane and $G$ is isomorphic to $\mathrm{P\Gamma L}(3,q)$ (for some prime-power $q$). In the more general case of a finite rank $2$ irreducible spherical building, also known as a \emph{generalized polygon}, the theorem of Fong and Seitz (1973) gave a classification of the \emph{Moufang} examples. A conjecture of Kantor, made in print in 1991, says that there are only two non-classical examples of flag-transitive generalized quadrangles up to duality. Recently, the authors made progress toward this conjecture by classifying those finite generalized quadrangles which have an automorphism group $G$ acting transitively on antiflags. In this paper, we take this classification much further by weakening the hypothesis to $G$ being transitive on ordered pairs of collinear points and ordered pairs of concurrent lines.