A classification of finite locally 2-transitive generalized quadrangles
Ostrom and Wagner (1959) proved that if the automorphism group $ G $ of a finite projective
plane $\pi $ acts $2 $-transitively on the points of $\pi $, then $\pi $ is isomorphic to the
Desarguesian projective plane and $ G $ is isomorphic to $\mathrm {P}\Gamma\mathrm
{L}(3, q) $(for some prime-power $ q $). In the more general case of a finite rank $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 …
plane $\pi $ acts $2 $-transitively on the points of $\pi $, then $\pi $ is isomorphic to the
Desarguesian projective plane and $ G $ is isomorphic to $\mathrm {P}\Gamma\mathrm
{L}(3, q) $(for some prime-power $ q $). In the more general case of a finite rank $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 …
Abstract
Ostrom and Wagner (1959) proved that if the automorphism group of a finite projective plane acts -transitively on the points of , then is isomorphic to the Desarguesian projective plane and is isomorphic to (for some prime-power ). In the more general case of a finite rank 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 acting transitively on antiflags. In this paper, we take this classification much further by weakening the hypothesis to being transitive on ordered pairs of collinear points and ordered pairs of concurrent lines. References
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