Nonexistence of generalized quadrangles admitting a point-primitive and line-primitive automorphism group with socle PSU(3,q), q≥3

A central problem in the study of generalized quadrangles is to classify finite generalized quadrangles satisfying certain symmetry conditions. It is known that an automorphism group of a finite thick generalized quadrangle S acting primitively on both the points and lines of S must be almost simple...

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Bibliographic Details
Published in:Journal of algebraic combinatorics Vol. 60; no. 3; pp. 871 - 898
Main Authors: Lu, Jianbing, Zhang, Yingnan, Zou, Hanlin
Format: Journal Article
Language:English
Published: New York Springer US 01.11.2024
Springer Nature B.V
Subjects:
Automorphisms
Combinatorics
Computer Science
Convex and Discrete Geometry
Geometry
Group Theory and Generalizations
Lattices
Mathematics
Mathematics and Statistics
Order
Ordered Algebraic Structures
Polygons
20B25
Generalized quadrangle
Primitive permutation group
51A10
51E12
05B25
20B15
ISSN:0925-9899, 1572-9192
Online Access:Get full text
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Summary:A central problem in the study of generalized quadrangles is to classify finite generalized quadrangles satisfying certain symmetry conditions. It is known that an automorphism group of a finite thick generalized quadrangle S acting primitively on both the points and lines of S must be almost simple. In this paper, we initiate the study of finite generalized quadrangles admitting a point-primitive and line-primitive automorphism group with socle being a unitary group. We develop a group-theoretic tool to prove that the socle of such a group cannot be PSU ( 3 , q ) with q ≥ 3 .
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ISSN:0925-9899
1572-9192
DOI:10.1007/s10801-024-01355-6