A classification of finite antiflag-transitive generalized quadrangles
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- by John Bamberg, Cai Heng Li and Eric Swartz PDF
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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.References
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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
- MR Author ID: 305568
- Email: cai.heng.li@uwa.edu.au
- Eric Swartz
- Affiliation: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187
- MR Author ID: 904405
- ORCID: 0000-0002-1590-1595
- Email: easwartz@wm.edu
- 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. - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 1551-1601
- MSC (2010): Primary 51E12, 20B05, 20B15, 20B25
- DOI: https://doi.org/10.1090/tran/6984
- MathSciNet review: 3739185