Innovations in Incidence Geometry — Algebraic, Topological and Combinatorial

Generalized quadrangles, Laguerre planes and shift planes of odd order

Günter F. Steinke and Markus Stroppel

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Abstract

We characterize the Miquelian Laguerre planes, and thus the classical orthogonal generalized quadrangles Q ( 4 , q ) , of odd order q by the existence of shift groups in affine derivations.

Article information

Source
Innov. Incidence Geom. Algebr. Topol. Comb., Volume 17, Number 1 (2019), 47-52.

Dates
Received: 11 October 2017
Revised: 20 November 2017
Accepted: 3 January 2018
First available in Project Euclid: 26 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.iig/1551206775

Digital Object Identifier
doi:10.2140/iig.2019.17.47

Mathematical Reviews number (MathSciNet)
MR3986547

Zentralblatt MATH identifier
06983418

Subjects
Primary: 51B15: Laguerre geometries 51E12: Generalized quadrangles, generalized polygons 51E15: Affine and projective planes
Secondary: 05B25: Finite geometries [See also 51D20, 51Exx] 51E25: Other finite nonlinear geometries

Keywords
generalized quadrangle orthogonal generalized quadrangle antiregular translation generalized quadrangle Laguerre plane Miquelian Laguerre plane translation plane shift plane shift group

Citation

Steinke, Günter F.; Stroppel, Markus. Generalized quadrangles, Laguerre planes and shift planes of odd order. Innov. Incidence Geom. Algebr. Topol. Comb. 17 (2019), no. 1, 47--52. doi:10.2140/iig.2019.17.47. https://projecteuclid.org/euclid.iig/1551206775


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References

  • R. Baer, “Projectivities with fixed points on every line of the plane”, Bull. Amer. Math. Soc. 52 (1946), 273–286.
  • L. R. A. Casse, J. A. Thas, and P. R. Wild, “$(q^n+1)$-sets of ${\rm PG}(3n-1,q)$, generalized quadrangles and Laguerre planes”, Simon Stevin 59:1 (1985), 21–42.
  • D. R. Hughes and F. C. Piper, Projective planes, Graduate Texts in Mathematics 6, Springer, 1973.
  • N. Knarr and M. Stroppel, “Polarities of shift planes”, Adv. Geom. 9:4 (2009), 577–603.
  • S. E. Payne and J. A. Thas, Finite generalized quadrangles, 2nd ed., European Mathematical Society (EMS), Zürich, 2009.
  • B. Spille and I. Pieper-Seier, “On strong isotopy of Dickson semifields and geometric implications”, Results Math. 33:3-4 (1998), 364–373.
  • G. F. Steinke, “On the structure of finite elation Laguerre planes”, J. Geom. 41:1-2 (1991), 162–179.
  • G. F. Steinke and M. J. Stroppel, “Finite elation Laguerre planes admitting a two-transitive group on their set of generators”, Innov. Incidence Geom. 13 (2013), 207–223.
  • G. F. Steinke and M. J. Stroppel, “On elation Laguerre planes with a two-transitive orbit on the set of generators”, Finite Fields Appl. 53 (2018), 64–84.
  • J. A. Thas, K. Thas, and H. Van Maldeghem, Translation generalized quadrangles, Series in Pure Mathematics 26, World Scientific Publishing Co., Hackensack, NJ, 2006.