A classification of finite antiflag-transitive generalized quadrangles

2015/08/14 by Bamberg, John, Li, Cai Heng, Swartz, Eric · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1508.03565

Abstract

A generalized quadrangle is a point-line 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 (1973), and we study a larger class of generalized quadrangles: the antiflag-transitive quadrangles. An antiflag of a generalized quadrangle is a non-incident point-line pair (P, ℓ), and we say that the generalized quadrangle Q is antiflag-transitive if the group of collineations is transitive on the set of all antiflags. We prove that if a finite thick generalized quadrangle Q is antiflag-transitive, then Q is either a classical generalized quadrangle or is the unique generalized quadrangle of order (3,5) or its dual.

Cited by

• Ealy's conjecture in odd characteristic
• Large maximal subgroups of almost simple classical groups

Related

• A classification of finite locally 2-transitive generalized quadrangles
• On generalized quadrangles with a point regular group of automorphisms
• No almost simple sporadic group acts primitively on the points of a generalised quadrangle
• On finite generalized quadrangles with PSL(2,q) as an automorphism group
• Generalized Quadrangles Of Order (s, s²). IV. Translations, Moufang And Fong-Seitz