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arXiv · 2403.00240

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

Jianbing Lu·Yingnan Zhang·Hanlin Zou
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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 $\mathcal{S}$ acting primitively on both the points and lines of $\mathcal{S}$ 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 $q\geq 3$.

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Jianbing Lu, Yingnan Zhang, Hanlin Zou. 2024-03-01. Nonexistence of generalized quadrangles admitting a point-primitive and line-primitive automorphism group with socle ${\rm PSU}(3,q)$, $q\geq 3$. https://arxiv.org/abs/2403.00240

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