A classification of finite antiflag-transitive generalized quadrangles

Abstract

Ostrom and Wagner (1959) proved that if the automorphism group G of a finite projective plane π acts 2-transitively on the points of π, then π is isomorphic to the Desarguesian projective plane and G is isomorphic to PΓL(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, 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 acting transitively on antiflags. In this paper, we take this classification much further by weakening the hypothesis to G being transitive on ordered pairs of collinear points and ordered pairs of concurrent lines.2010 Mathematics Subject Classification. Primary 51E12, 20B05, 20B15, 20B25. 1 2 JOHN BAMBERG, CAI HENG LI, AND ERIC SWARTZ Conjecture 1.1 (W. M. Kantor 1991). If Q is a finite flag-transitive generalized quadrangle and Q is not a classical generalized quadrangle, then (up to duality) Q is the unique generalized quadrangle of order (3, 5) or the generalized quadrangle of order (15, 17) arising from the Lunelli-Sce hyperoval.Finite generalized polygons satisfying stronger symmetry assumptions, such as the Moufang condition [15] or distance-transitivity [12,28], have been classified. The current state-of-the-art for generalized quadrangles is the classification of antiflag-transitive finite generalized quadrangles in [5], where it was shown that, up to duality, the only nonclassical antiflag-transitive generalized quadrangle is the unique generalized quadrangle of order (3,5). (An antiflag is a non-incident point-line pair.) Notice that by the GQ Axiom, antiflag-transitivity implies flag-transitivity for a generalized quadrangle.The aim of this paper is to provide further progress toward Conjecture 1.1. We study finite generalized quadrangles with a strictly weaker local symmetry condition, local 2transitivity. If G is a subgroup of collineations of a finite generalized quadrangle Q that is transitive both on pairs of collinear points and pairs of concurrent lines, then Q is said to be a locally (G, 2)-transitive generalized quadrangle.Our main result is a complete classification of thick locally 2-transitive generalized quadrangles, the proof of which relies on the Classification of Finite Simple Groups (CFSG) [20].Theorem 1.2. If Q is a thick finite locally (G, 2)-transitive generalized quadrangle and Q is not a classical generalized quadrangle, then (up to duality) Q is the unique generalized quadrangle of order (3, 5).An equivalent definition of a locally 2-transitive generalized quadrangle is that it has an incidence graph that is locally 2-arc-transitive; see Section 2 for details. Since an equivalent definition of an antiflag-transitive generalized quadrangl...

Subgroups of Classical Groups that are Transitive on Subspaces

“…Thus when q is even, (H) τ is given by Theorem 7.2(e) and when q is odd it is given by Theorem 7.1(b). By (6),…”

“…by (6). Letting q = p f we see that |H τ | must be divisible by a primitive prime divisor r of p 4f − 1 (one always exists).…”

“…It follows that the subgroup Sp 2a (q b ) is also transitive on N + 2m , and hence if Sp 2a (q b ) H then H is transitive on U and we have listed this in Table 7. 6. Suppose now that Sp 2a (q b ) H. Then we have a factorisation B = HB U .…”

“…Thus if G is one of these three groups and p is a point of the corresponding generalised quadrangle Q then G p has two orbits on the remaining points: the points collinear with p and the points not collinear with p. Since each pair of points lies on at most one line, it follows that G p acts transitively on the set of lines of Q incident with p and so G acts flag-transitively on Q. Kantor [21] conjectured that, up to duality, the only flag-transitive generalised quadrangles are the classical ones and the unique generalised quadrangles of order (3,5) and (15,17). This conjecture is still wide-open but recently, Bamberg, Li and Swartz [6] showed that, up to duality, the only antiflag-transitive generalised quadrangles are the classical ones and the unique generalised quadrangle of order (3,5). Theorem 9.3.…”

“…By [31,Table 2], Sz(q) is transitive on U and so Sz(q) H is listed in Table 7. 6. Suppose now that Sz(q) H. Then we have a factorisation B = HB U .…”