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Every generalized quadrangle of order 5 having a regular point is symplectic

Bart De Bruyn (UGent)
(2015) EXPERIMENTAL MATHEMATICS. 24(1). p.45-52
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Abstract
For many years now, one of the most important open problems in the theory of generalized quadrangles has been whether other classes of generalized quadrangles exist besides those that are currently known. This paper reports on an unsuccessful attempt to construct a new generalized quadrangle. As a byproduct of our attempt, however, we obtain the following new characterization result: every generalized quadrangle of order 5 that has at least one regular point is isomorphic to the quadrangle W(5) arising from a symplectic polarity of PG(3, 5). During the classification process, we used the computer algebra system GAP to perform certain computations or to search for an optimal strategy for the proof.
Keywords
generalized quadrangle, regular point, (generalized) admissible triple, affine plane, COORDINATIZATION, SPREAD

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Citation

Please use this url to cite or link to this publication:

Chicago
De Bruyn, Bart. 2015. “Every Generalized Quadrangle of Order 5 Having a Regular Point Is Symplectic.” Experimental Mathematics 24 (1): 45–52.
APA
De Bruyn, B. (2015). Every generalized quadrangle of order 5 having a regular point is symplectic. EXPERIMENTAL MATHEMATICS, 24(1), 45–52.
Vancouver
1.
De Bruyn B. Every generalized quadrangle of order 5 having a regular point is symplectic. EXPERIMENTAL MATHEMATICS. 2015;24(1):45–52.
MLA
De Bruyn, Bart. “Every Generalized Quadrangle of Order 5 Having a Regular Point Is Symplectic.” EXPERIMENTAL MATHEMATICS 24.1 (2015): 45–52. Print.
@article{7140152,
  abstract     = {For many years now, one of the most important open problems in the theory of generalized quadrangles has been whether other classes of generalized quadrangles exist besides those that are currently known. This paper reports on an unsuccessful attempt to construct a new generalized quadrangle. As a byproduct of our attempt, however, we obtain the following new characterization result: every generalized quadrangle of order 5 that has at least one regular point is isomorphic to the quadrangle W(5) arising from a symplectic polarity of PG(3, 5). During the classification process, we used the computer algebra system GAP to perform certain computations or to search for an optimal strategy for the proof.},
  author       = {De Bruyn, Bart},
  issn         = {1058-6458},
  journal      = {EXPERIMENTAL MATHEMATICS},
  keyword      = {generalized quadrangle,regular point,(generalized) admissible triple,affine plane,COORDINATIZATION,SPREAD},
  language     = {eng},
  number       = {1},
  pages        = {45--52},
  title        = {Every generalized quadrangle of order 5 having a regular point is symplectic},
  url          = {http://dx.doi.org/10.1080/10586458.2014.952051},
  volume       = {24},
  year         = {2015},
}

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