Slanted symplectic quadrangles

Grundhöfer, Theo; Joswig, Michael; Stroppel, Markus

doi:10.1007/bf01610617

Cited by 16 publications

(14 citation statements)

References 11 publications

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“…The only 4-transitive groups of degree 10 are A 10 and S 10 (see [25]), and we claim that G cannot induce A 10 (nor S 10 , therefore) on O. Now, a collineation of PG (8,2) [20, 3.2.6]), and Grunhöfer et al [12] have shown that the automorphism group of the Payne derivation of W (3, 8) is a point stabilizer in P Sp (4,8). This group therefore has the form 2 9 : (7 · PSL(2, 8) · 3), and hence, by divisibility, cannot contain A 10 as a subgroup.…”

Section: Proof Of Proposition 12mentioning

confidence: 72%

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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Section: Proof Of Proposition 12mentioning

confidence: 72%

Generalized Quadrangles and Transitive Pseudo‐Hyperovals

Bamberg

Glasby

Popiel

et al. 2014

J of Combinatorial Designs

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“…Bichara, Mazzocca and Somma showed in [4] that for K, K ′ hyperovals, T * 2 (K) ∼ = T * 2 (K ′ ) if and only if K and K ′ are PΓL-equivalent. The answer to (Q) is given in the affirmative when K is a regular hyperoval in PG (2, q) in [11]. When K is a Buekenhout-Metz unital, question (Q) is answered by De Winter in [10]; in this case, the linear representation T * 2 (K) is a semipartial geometry.…”

Section: Incidence Is Naturalmentioning

confidence: 99%

Linear representations of subgeometries

Winter

Rottey

Voorde

2014

Des. Codes Cryptogr.

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The linear representation T∗n(K) of a point set K in a hyperplane of PG(n+1,q) is a point-line geometry embedded in this projective space. In this paper, we will determine the isomorphisms between two linear representations T∗n(K) and T∗n(K′), under a few conditions on K and K′. First, we prove that an isomorphism between T∗n(K) and T∗n(K′) is induced by an isomorphism between the two linear representations T∗n(K) and T∗n(K′) of their closures K and K′. This allows us to focus on the automorphism group of a linear representation T∗n(S) of a subgeometry S≅PG(n,q) embedded in a hyperplane of the projective space PG(n+1,qt). To this end we introduce a geometry X(n,t,q) and determine its automorphism group. The geometry X(n,t,q) is a straightforward generalization of Hn+2q which is known to be isomorphic to the linear representation of a Baer subgeometry. By providing an elegant algebraic description of X(n,t,q) as a coset geometry we extend this result and prove that X(n,t,q) and T∗n(S) are isomorphic. Finally, we compare the full automorphism group of T∗n(S) with the “natural” group of automorphisms that is induced by the collineation group of its ambient space

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“…We will give a definition of Payne derivation as described in [STdW09]. The same procedure is called slanting in [GJS94]. We write p ∼ r for collinear points p and r. Then we define…”

Section: Payne Derivationmentioning

confidence: 99%

A geometric construction of panel-regular lattices for buildings of types Ã₂and C̃₂

Essert

2013

Algebr. Geom. Topol.

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Using Singer polygons, we construct locally finite affine buildings of types Ã2 and C2 which admit uniform lattices acting regularly on panels. This construction produces very explicit descriptions of these buildings as well as very short presentations of the lattices. All but one of the C2 -buildings are necessarily exotic. To the knowledge of the author, these are the first presentations of lattices in buildings of type C2 . Integral and rational group homology for the lattices is also calculated.

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Slanted symplectic quadrangles

Cited by 16 publications

References 11 publications

Generalized Quadrangles and Transitive Pseudo‐Hyperovals

Generalized Quadrangles and Transitive Pseudo‐Hyperovals

Linear representations of subgeometries

A geometric construction of panel-regular lattices for buildings of types Ã₂and C̃₂

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Slanted symplectic quadrangles

Cited by 16 publications

References 11 publications

Generalized Quadrangles and Transitive Pseudo‐Hyperovals

Generalized Quadrangles and Transitive Pseudo‐Hyperovals

Linear representations of subgeometries

A geometric construction of panel-regular lattices for buildings of types Ã2and C̃2

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A geometric construction of panel-regular lattices for buildings of types Ã₂and C̃₂