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

Vishnuram Arumugam, John Bamberg, Michael Giudici

TL;DR

The paper studies finite thick generalised quadrangles under automorphism groups acting primitively on both points and lines. It proves that no almost simple group with socle $Sz(2^{2m+1})$ or $Ree(3^{2m+1})$ can realize point- and line-primitivity simultaneously, by exploiting the maximal-subgroup structure of these rank-1 groups, analyzing involution and order-3 element behavior, and applying fixed-point count techniques in primitive actions. The approach extends Morgan–Popiel’s results for higher-rank polygons to the quadrangle setting, narrowing the landscape of possible socles in such symmetric actions. These results refine the understanding of symmetry constraints for finite generalized quadrangles and guide further classification of groups acting primitively on both points and lines.

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.

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

TL;DR

The paper studies finite thick generalised quadrangles under automorphism groups acting primitively on both points and lines. It proves that no almost simple group with socle or can realize point- and line-primitivity simultaneously, by exploiting the maximal-subgroup structure of these rank-1 groups, analyzing involution and order-3 element behavior, and applying fixed-point count techniques in primitive actions. The approach extends Morgan–Popiel’s results for higher-rank polygons to the quadrangle setting, narrowing the landscape of possible socles in such symmetric actions. These results refine the understanding of symmetry constraints for finite generalized quadrangles and guide further classification of groups acting primitively on both points and lines.

Abstract

Suppose we have a finite thick generalised quadrangle whose automorphism group acts primitively on both the set of points and the set of lines. Then must be almost simple. In this paper, we show that cannot be isomorphic to or where is a positive integer.
Paper Structure (13 sections, 16 theorems, 11 equations, 2 tables)

This paper contains 13 sections, 16 theorems, 11 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Incidence Geometry
  4. Group Theory
  5. Notation
  6. Elementary Results
  7. Permutation Groups
  8. Suzuki Groups
  9. Structure of Suzuki Groups
  10. Suzuki Groups Acting on Generalised Quadrangles
  11. Ree Groups
  12. Structure of Ree Groups
  13. Ree Groups Acting on Generalised Quadrangles

Key Result

Lemma 2.1

Let $G$ be a group acting on a generalised quadrangle $\Gamma$ and suppose $g \in G$. Let $\Gamma_g = (\mathcal{P}_g,\mathcal{L}_g,\mathcal{I}_g)$ be the fixed substructure of $g$. If $|\mathcal{P}_g| \geqslant 2$, $|\mathcal{L}_g| \geqslant 2$ and $\Gamma_g$ admits an automorphism group $H$ that is

Theorems & Definitions (16)

  • Lemma 2.1: fenglu
  • Lemma 2.2
  • Theorem 2.3: Schur-Zassenhaus Theorem
  • Corollary 2.4
  • Corollary 2.5
  • Lemma 2.6: Formula for the Number of Fixed Points fixpts
  • Lemma 2.7: luzhangzou
  • Theorem 3.1: Maximal Subgroups of a Suzuki Group
  • Lemma 3.2: Conjugacy Class of Involutions
  • Lemma 3.3: Centraliser of an Involution
  • ...and 6 more