The Lunelli-Sce hyperoval inPG(2, 16)

Brown, Julia M. Nowlin; Cherowitzo, William E.

doi:10.1007/bf01237471

Cited by 8 publications

(13 citation statements)

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“…If both G B and G C are isomorphic to M 11 , then, since |G : G P | = (s + 1)(11s + 1) > 12 = (t + 1) = |G : G C |, we must have G P < G C . The only proper subgroup of M 11 with a 2-transitive action on 12 elements is PSL(2, 11), so we conclude that G P ∼ = PSL (2,11), and so (s + 1)(11s + 1) = |P| = |G : G P | = 144, which is a contradiction to s ∈ N. If G C = PSL(2, 11), then, since PSL(2, 11) has no proper subgroups with a 2-transitive action on 12 elements, we conclude that G P = G C and reach a contradiction as in the previous case. Finally, if G C = PGL(2, 11), then G = M 12 .2 and, since s > 2, we again conclude that G P = G C and reach a contradiction as in the previous cases.…”

Section: Imprimitive On Both Points and Linesmentioning

confidence: 76%

“…If G = soc(G) = M 11 , then t + 1 = 11, G B = M 10 , and G C = PSL (2,11). Since PSL (2,11) has no proper subgroups with a 2-transitive action on 11 elements, we conclude that G P = G C . However, this means that (s + 1)(10s + 1) = |P| = |G : G P | = 12, a contradiction.…”

Section: Imprimitive On Both Points and Linesmentioning

confidence: 87%

“…Since r ∼ = ±3 (mod 8) and r > 5, this means that T ∼ = PSL 2 (11). Moreover, if k 5, then we have |T | = 660 < 144 k k−1 144 5 4 < 499, a contradiction, and hence k = 4.…”

Section: Proofmentioning

confidence: 93%

“…Remaining possibilities for (soc(G), G C ). soc(G) t + 1 Possible G C PSL(2, 11) 11 ∅ M 11 11 PSL(2, 11) M 11 12 ∅ M 12 12 M 11 , PSL(2, 11), PGL(2, 11) A 7 15 ∅ M 22 22 ∅ M 23 23 ∅ M 24 24 PSL(2, 23)…”

Section: Imprimitive On Both Points and Linesmentioning

confidence: 99%

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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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Section: Imprimitive On Both Points and Linesmentioning

confidence: 76%

Section: Imprimitive On Both Points and Linesmentioning

confidence: 87%

“…Since r ∼ = ±3 (mod 8) and r > 5, this means that T ∼ = PSL 2 (11). Moreover, if k 5, then we have |T | = 660 < 144 k k−1 144 5 4 < 499, a contradiction, and hence k = 4.…”

Section: Proofmentioning

confidence: 93%

Section: Imprimitive On Both Points and Linesmentioning

confidence: 99%

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A classification of finite antiflag-transitive generalized quadrangles

Bamberg

Swartz

2017

Trans. Amer. Math. Soc.

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“…Therefore, there are two related ovals (consequently two Niho bent functions) up to equivalence, obtained from functions g(u) = 1 and g ′ (u) = 1 + u 4 + ū4 +u 5 + ū5 +u 8 + ū8 . The automorphism group of the Lunelli-Sce hyperoval is transitive on the points of the hyperoval [10,35]. Therefore, it determines only one oval (consequently one Niho bent function) with the function g ′′ (u) = 1+u 5 + ū5 (the Lunelli-Sce hyperoval is the first non-trivial member of the Subiaco [10,22] and the Adelaide families [24]).…”

Section: Niho Bent Functions In Small Dimensionsmentioning

confidence: 99%

Equivalence classes of Niho bent functions

Abdukhalikov

2019

Preprint

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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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The Lunelli-Sce hyperoval inPG(2, 16)

Cited by 8 publications

References 7 publications

A classification of finite antiflag-transitive generalized quadrangles

A classification of finite antiflag-transitive generalized quadrangles

Equivalence classes of Niho bent functions

Generalized Quadrangles and Transitive Pseudo‐Hyperovals

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