A classification of finite locally $2$-transitive generalized quadrangles
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- by John Bamberg, Cai Heng Li and Eric Swartz PDF
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Abstract:
Ostrom and Wagner (1959) proved that if the automorphism group $G$ of a finite projective plane $\pi$ acts $2$-transitively on the points of $\pi$, then $\pi$ is isomorphic to the Desarguesian projective plane and $G$ is isomorphic to $\mathrm {P} \Gamma \mathrm {L}(3,q)$ (for some prime-power $q$). In the more general case of a finite rank $2$ irreducible spherical building, also known as a generalized polygon, the theorem of Fong and Seitz (1973) gave a classification of the Moufang examples. A conjecture of Kantor, made in print in 1991, says that there are only two non-classical examples of flag-transitive generalized quadrangles up to duality. Recently, the authors made progress toward this conjecture by classifying those finite generalized quadrangles which have an automorphism group $G$ acting transitively on antiflags. In this paper, we take this classification much further by weakening the hypothesis to $G$ being transitive on ordered pairs of collinear points and ordered pairs of concurrent lines.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, W.A. 6009, Australia
- MR Author ID: 670428
- ORCID: 0000-0001-7347-8687
- Email: john.bamberg@uwa.edu.au
- Cai Heng Li
- Affiliation: SUSTech International Center for Mathematics, Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong 518055, People’s Republic of China
- Email: lich@sustech.edu.cn
- Eric Swartz
- Affiliation: Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23187
- MR Author ID: 904405
- ORCID: 0000-0002-1590-1595
- Email: easwartz@wm.edu
- Received by editor(s): July 6, 2019
- Received by editor(s) in revised form: January 17, 2020, February 26, 2020, and April 17, 2020
- Published electronically: November 25, 2020
- Additional Notes: The second author acknowledges the support of NSFC grants 11771200 and 11231008.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 1535-1578
- MSC (2020): Primary 51E12, 20B05, 20B15, 20B25
- DOI: https://doi.org/10.1090/tran/8236
- MathSciNet review: 4216717