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Abstract
A generalized quadrangle is a pointline incidence geometry Q such that: (i) any two points lie on at most one line, and (ii) given a line ℓ and a point P not incident with ℓ, there is a unique point of ℓ collinear with P. The finite Moufang generalized quadrangles were classified by Fong and Seitz [Invent. Math. 21 (1973), 157; Invent. Math. 24 (1974), 191239], and we study a larger class of generalized quadrangles: the antiflagtransitive quadrangles. An antiflag of a generalized quadrangle is a nonincident pointline pair (P, ℓ), and we say that the generalized quadrangle Q is antiflagtransitive if the group of collineations is transitive on the set of all antiflags. We prove that if a finite thick generalized quadrangle Q is antiflagtransitive, then Q is either a classical generalized quadrangle or is the unique generalized quadrangle of order (3, 5) or its dual. Our approach uses the theory of locally sarctransitive graphs developed by Giudici, Li, and Praeger [Trans. Amer. Math. Soc. 356 (2004), 291317] to characterize antiflagtransitive generalized quadrangles and then the work of Alavi and Burness [J. Algebra 421 (2015), 187233] on “large” subgroups of simple groups of Lie type to fully classify them.
Original language  English 

Pages (fromto)  15511601 
Number of pages  51 
Journal  Transactions of the American Mathematical Society 
Volume  370 
Issue number  3 
DOIs  
Publication status  Published  1 Mar 2018 
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 2 Finished


Cayley Graphs & their Associated Geometric & Combinatorial Objects
Li, C. & Bamberg, J.
1/01/12 → 30/06/17
Project: Research