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

2024/03/01 by Lu, Jianbing, Zhang, Yingnan, Zou, Hanlin
#05B25 #20B15 #20B25 #51A10 #51E12 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2403.00240

Abstract

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 \rm PSU(3,q) with q≥ 3.

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