Translation Generalized Quadrangles In Even Characteristic

In this paper, we first introduce new objects called “translation generalized ovals” and “translation generalized ovoids”, and make a thorough study of these objects. We then obtain numerous new characterizations of the \( T_{2} {\left( {\user1{\mathcal{O}}} \right)} \) of Tits and the classical generalized quadrangle \( {\user1{\mathcal{Q}}}{\left( {4,q} \right)} \) in even characteristic, including the complete classification of 2-transitive generalized ovals for the even case. Next, we prove a new strong characterization theorem for the \( T_{3} {\left( {\user1{\mathcal{O}}} \right)} \) of Tits. As a corollary, we obtain a purely geometric proof of a theorem of Johnson on semifield flocks.

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Correspondence to Joseph A. Thas.

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* The second author is a Postdoctoral Fellow of the Fund for Scientific Research—Flanders (Belgium).

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Thas, J.A., Thas, K. Translation Generalized Quadrangles In Even Characteristic. Combinatorica 26, 709–732 (2006). https://doi.org/10.1007/s00493-006-0038-6

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Mathematics Subject Classification (2000):

  • 05B25
  • 05E20
  • 20B25
  • 51E12
  • 51E20