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A classification of finite antiflag-transitive generalized quadrangles


Authors: John Bamberg, Cai Heng Li and Eric Swartz
Journal: Trans. Amer. Math. Soc. 370 (2018), 1551-1601
MSC (2010): Primary 51E12, 20B05, 20B15, 20B25
DOI: https://doi.org/10.1090/tran/6984
Published electronically: August 3, 2017
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Abstract: A generalized quadrangle is a point-line incidence geometry $ \mathcal {Q}$ such that: (i) any two points lie on at most one line, and (ii) given a line $ \ell $ and a point $ P$ not incident with $ \ell $, there is a unique point of $ \ell $ collinear with $ P$. The finite Moufang generalized quadrangles were classified by Fong and Seitz [Invent. Math. 21 (1973), 1-57; Invent. Math. 24 (1974), 191-239], and we study a larger class of generalized quadrangles: the antiflag-transitive quadrangles. An antiflag of a generalized quadrangle is a nonincident point-line pair $ (P, \ell )$, and we say that the generalized quadrangle $ \mathcal {Q}$ is antiflag-transitive if the group of collineations is transitive on the set of all antiflags. We prove that if a finite thick generalized quadrangle $ \mathcal {Q}$ is antiflag-transitive, then $ \mathcal {Q}$ is either a classical generalized quadrangle or is the unique generalized quadrangle of order $ (3,5)$ or its dual. Our approach uses the theory of locally $ s$-arc-transitive graphs developed by Giudici, Li, and Praeger [Trans. Amer. Math. Soc. 356 (2004), 291-317] to characterize antiflag-transitive generalized quadrangles and then the work of Alavi and Burness [J. Algebra 421 (2015), 187-233] on ``large'' subgroups of simple groups of Lie type to fully classify them.


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Additional Information

John Bamberg
Affiliation: Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia
Email: john.bamberg@uwa.edu.au

Cai Heng Li
Affiliation: Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia
Email: cai.heng.li@uwa.edu.au

Eric Swartz
Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187
Email: easwartz@wm.edu

DOI: https://doi.org/10.1090/tran/6984
Received by editor(s): April 16, 2016
Received by editor(s) in revised form: May 14, 2016
Published electronically: August 3, 2017
Additional Notes: The first author was supported by an Australian Research Council Future Fellowship (FT120100036).
This paper forms part of an Australian Research Council Discovery Project (DP120101336) that supported the third author during his time at The University of Western Australia.
Article copyright: © Copyright 2017 American Mathematical Society

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