Low rank groups of Lie type acting point and line-primitively on finite generalised quadrangles

TL;DR

This paper investigates the automorphism groups of finite thick generalized quadrangles, proving restrictions on their socle types, specifically excluding certain groups of Lie type from acting primitively on points and lines.

Contribution

It establishes that the socle of automorphism groups acting point and line-primitively cannot be isomorphic to certain Suzuki or Ree groups, advancing understanding of symmetry in finite geometries.

Findings

Soc(G) cannot be isomorphic to Sz(2^{2m+1})

Soc(G) cannot be isomorphic to Ree(3^{2m+1})

Automorphism groups are almost simple under these conditions

Abstract

Suppose we have a finite thick generalised quadrangle whose automorphism group $G$ acts primitively on both the set of points and the set of lines. Then $G$ must be almost simple. In this paper, we show that $\operatorname{soc}(G)$ cannot be isomorphic to $\operatorname{Sz}(2^{2m+1})$ or $\operatorname{Ree}(3^{2m+1})$ where $m$ is a positive integer.