Generalised quadrangles with a group of automorphisms acting primitively on points and lines

Bamberg, John; Giudici, Michael; Morris, Joy; Royle, Gordon F.; Spiga, Pablo

doi:10.1016/j.jcta.2012.04.005

Cited by 19 publications

(107 citation statements)

References 20 publications

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“…Theorem 2.1). For some results on flag-transitive generalized quadrangles, we refer to Bamberg et al [5]; for flag-transitive generalized hexagons, we refer to Schneider and Van Maldeghem [30].…”

Section: Lemma 54 Let π Be a Projective Plane Of Order Q With Q Evementioning

confidence: 99%

Distance-regular Cayley graphs with least eigenvalue $$-2$$ - 2

Abdollahi

Dam

Jazaeri

2016

Des. Codes Cryptogr.

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We classify the distance-regular Cayley graphs with least eigenvalue −2 and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes. In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of distance-regular Cayley line graphs of incidence graphs of generalized polygons.

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Section: Lemma 54 Let π Be a Projective Plane Of Order Q With Q Evementioning

confidence: 99%

Distance-regular Cayley graphs with least eigenvalue $$-2$$ - 2

Abdollahi

Dam

Jazaeri

2016

Des. Codes Cryptogr.

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“…Note that the regular hyperoval of PG(2, 2) does not appear in Corollary 1.5 because it does not yield a thick generalized quadrangle (see Section 5). Note also that if H is the regular hyperoval of PG (2,4), then T * 2 (H) is the unique generalized quadrangle of order (3,5) [20, 6.2.4].…”

Section: Corollary 15mentioning

confidence: 99%

“…By contrast, C acts regularly on a conic in PG (2,8), and fixes the nucleus. Identifying the vectors of V (3,8) with those of V (9, 2), we see that the fixed-point space of C is a 3-dimensional F 2 -subspace. This contradiction proves our claim.…”

Section: Remark 33mentioning

confidence: 99%

Generalized Quadrangles and Transitive Pseudo‐Hyperovals

Bamberg

Glasby

Popiel

et al. 2014

J of Combinatorial Designs

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A pseudo-hyperoval of a projective space PG(3n − 1, q), q even, is a set of q n + 2 subspaces of dimension n − 1 such that any three span the whole space. We prove that a pseudohyperoval with an irreducible transitive stabilizer is elementary. We then deduce from this result a classification of the thick generalized quadrangles Q that admit a point-primitive, linetransitive automorphism group with a point-regular abelian normal subgroup. Specifically, we show that Q is flag-transitive and isomorphic to T * 2 (H), where H is either the regular hyperoval of PG(2, 4) or the Lunelli-Sce hyperoval of PG(2, 16). (i) H acts transitively on the lines of Q;(ii) H acts transitively on the flags of Q; (iii) H acts transitively on the pseudo-hyperoval {U 1 , U 2 , . . . , U t+1 }, by conjugation.Theorem 1.1 is proved in Section 2. It implies that a classification of transitive pseudohyperovals would yield a classification of the generalized quadrangles that admit a line-transitive automorphism group with a point-regular abelian normal subgroup, and, moreover, that such generalized quadrangles are, in fact, flag-transitive. By a result of J. A. Thas [28, §4.5], if a projective space PG(3n − 1, q) contains a pseudo-hyperoval, then q = 2 f for some positive integer f . For small values of the product nf , we appeal to some existing results to classify the transitive pseudo-hyperovals of PG(3n − 1, 2 f ), Journal of Combinatorial Designs

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“…Assume Hypothesis 4.1. If G acts with type HA on P, then Q is the unique generalized quadrangle of order (3,5).…”

Section: Quasiprimitive On Points But Not Linesmentioning

confidence: 99%

“…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.)…”

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confidence: 99%

A classification of finite antiflag-transitive generalized quadrangles

Bamberg

Swartz

2017

Trans. Amer. Math. Soc.

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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...

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Generalised quadrangles with a group of automorphisms acting primitively on points and lines

Abstract: We show that if G is a group of automorphisms of a thick finite generalised quadrangle Q acting primitively on both the points and lines of Q, then G is almost simple. Moreover, if G is also flag-transitive then G is of Lie type.

Cited by 19 publications

References 20 publications

Distance-regular Cayley graphs with least eigenvalue $$-2$$ - 2

Distance-regular Cayley graphs with least eigenvalue $$-2$$ - 2

Generalized Quadrangles and Transitive Pseudo‐Hyperovals

A classification of finite antiflag-transitive generalized quadrangles

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