On finite generalized quadrangles with $\mathrm{PSL}(2,q)$ as an automorphism group

TL;DR

This paper proves that for a finite thick generalized quadrangle with an automorphism group acting primitively on points and lines, if the socle is PSL(2,q) with q≥4, then q=9 and the quadrangle is uniquely of order 2.

Contribution

It establishes a classification result linking the socle PSL(2,q) to a unique generalized quadrangle of order 2 when q=9.

Findings

q=9 is the only case with socle PSL(2,q) for q≥4

The generalized quadrangle in this case is unique of order 2

Automorphism group acts primitively on points and lines

Abstract

Let $\mathcal{S}$ be a finite thick generalized quadrangle, and suppose that $G$ is an automorphism group of $\mathcal{S}$. If $G$ acts primitively on both the points and lines of $\mathcal{S}$, then it is known that $G$ must be almost simple. In this paper, we show that if the socle of $G$ is $\mathrm{PSL}(2,q)$ with $q\geq4$, then $q=9$ and $\mathcal{S}$ is the unique generalized quadrangle of order $2$.