Generalised quadrangles with a group of automorphisms acting primitively on points and lines

TL;DR

This paper proves that for thick finite generalised quadrangles with automorphism groups acting primitively on points and lines, the automorphism group is almost simple, and of Lie type if also flag-transitive.

Contribution

It establishes the structure of automorphism groups of generalised quadrangles under primitive and flag-transitive actions, showing they are almost simple and of Lie type respectively.

Findings

Automorphism group is almost simple under primitive actions.

Flag-transitivity implies automorphism group is of Lie type.

Provides classification constraints for symmetries of generalised quadrangles.

Abstract

We show that if G is a group of automorphisms of a thick finite generalised quadrangle Q acting primitively on both the points and lines of Q, then G is almost simple. Moreover, if G is also flag-transitive then G is of Lie type.