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Annals of Combinatorics

, Volume 20, Issue 3, pp 433–452 | Cite as

On Semi-Finite Hexagons of Order (2, t) Containing a Subhexagon

  • Anurag Bishnoi
  • Bart De BruynEmail author
Article

Abstract

The research in this paper was motivated by one of the most important open problems in the theory of generalized polygons, namely the existence problem for semi–finite thick generalized polygons. We show here that no semi–finite generalized hexagon of order (2, t) can have a subhexagon H of order 2. Such a subhexagon is necessarily isomorphic to the split Cayley generalized hexagon H(2) or its point–line dual H D (2). In fact, the employed techniques allow us to prove a stronger result. We show that every near hexagon \({\mathcal{S}}\) of order (2, t) which contains a generalized hexagon H of order 2 as an isometrically embedded subgeometry must be finite. Moreover, if \({H \cong H^{D}}\)(2) then \({\mathcal{S}}\) must also be a generalized hexagon, and consequently isomorphic to either H D (2) or the dual twisted triality hexagon T(2, 8).

Keywords

generalized hexagon near hexagon valuation 

Mathematics Subject Classification

51E12 05B25 

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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsGhent UniversityGentBelgium

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