Abstract
If a GQ S′ of order (s, s) is contained in a GQ S of order (s, s 2) as a subquadrangle, then for each point X of S\S′ the set of points \(O_x \) of S′ collinear with X form an ovoid of S′. Thas and Payne proved that if S′= \(Q\)(4,q),q even, and \(O_x \) is an elliptic quadric for each X∈S\S′,thenS≅ \(Q\)(5,q). In this paper we provide a single proof for the q odd and q even cases by establishing a link between the geometry involved and the first cohomology group of a related simplicial complex.
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Brown, M.R. A Characterisation of the Generalized Quadrangle Q (5, q) Using Cohomology. Journal of Algebraic Combinatorics 15, 107–125 (2002). https://doi.org/10.1023/A:1013812619953
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DOI: https://doi.org/10.1023/A:1013812619953