On generalized quadrangles with a point regular group of automorphisms

TL;DR

This paper investigates the symmetry groups of generalized quadrangles, characterizing those with automorphism groups acting regularly on points and lines, and establishing restrictions on their group structures.

Contribution

It provides a characterization of generalized quadrangles with point- and line-regular automorphism groups and shows such structures do not admit polarities or certain nonabelian normal subgroups.

Findings

Generalized quadrangles with point-regular automorphism groups are characterized.

Such quadrangles do not admit polarities.

Groups acting regularly on specific quadrangles lack nonabelian minimal normal subgroups.

Abstract

A generalized quadrangle is a point-line incidence geometry such that any two points lie on at most one line and, given a line $\ell$ and a point $P$ not incident with $\ell$, there is a unique point of $\ell$ collinear with $P$. We study the structure of groups acting regularly on the point set of a generalized quadrangle. In particular, we provide a characterization of the generalized quadrangles with a group of automorphisms acting regularly on both the point set and the line set and show that such a thick generalized quadrangle does not admit a polarity. Moreover, we prove that a group $G$ acting regularly on the point set of a generalized quadrangle of order $(u^2, u^3)$ or $(s,s)$, where $s$ is odd and $s+1$ is coprime to $3$, cannot have any nonabelian minimal normal subgroups.