A classification of finite locally 2-transitive generalized quadrangles

Abstract

Ostrom and Wagner (1959) proved that if the automorphism group G G of a finite projective plane π \pi acts 2 2 -transitively on the points of π \pi , then π \pi is isomorphic to the Desarguesian projective plane and G G is isomorphic to P Γ L ( 3 , q ) \mathrm {P} \Gamma \mathrm {L}(3,q) (for some prime-power q q ). In the more general case of a finite rank 2 2 irreducible spherical building, also known as a generalized polygon, the theorem of Fong and Seitz (1973) gave a classification of the Moufang examples. A conjecture of Kantor, made in print in 1991, says that there are only two non-classical examples of flag-transitive generalized quadrangles up to duality. Recently, the authors made progress toward this conjecture by classifying those finite generalized quadrangles which have an automorphism group G G acting transitively on antiflags. In this paper, we take this classification much further by weakening the hypothesis to G G being transitive on ordered pairs of collinear points and ordered pairs of concurrent lines.