Symplectic Quadrangle Research Articles

The point regular automorphism groups of the Payne derived quadrangle of W(q)

In this paper, we completely determine the point regular automorphism groups of the Payne derived quadrangle of the symplectic quadrangle W(q), q odd. As a corollary, we show that the finite groups that act regularly on the points of a finite generalized quadrangle can have unbounded nilpotency class.

Point regular groups of automorphisms of generalised quadrangles

We study the point regular groups of automorphisms of some of the known generalised quadrangles. In particular we determine all point regular groups of automorphisms of the thick classical generalised quadrangles. We also construct point regular groups of automorphisms of the generalised quadrangle of order ( q − 1 , q + 1 ) obtained by Payne derivation from the classical symplectic quadrangle W ( 3 , q ) . For q = p f with f ⩾ 2 we obtain at least two nonisomorphic groups when p ⩾ 5 and at least three nonisomorphic groups when p = 2 or 3. Our groups include nonabelian 2-groups, groups of exponent 9 and nonspecial p-groups. We also enumerate all point regular groups of automorphisms of some small generalised quadrangles.

Linearly small elation quadrangles

We show that each elation generalized quadrangle with parameters (p, p), where p is a prime, is isomorphic to the symplectic quadrangle W(p) or its dual Q(4, p). Our results cover the more general case of linearly small elation generalized quadrangles. In particular, we obtain a characterization of the symplectic quadrangle over the field of complex numbers among compact connected quadrangles. We prove that every root elation quadrangle (Q, c, H F ) is a skew translation quadrangle.

A coordinatization structure for generalized quadrangles with a regular spread

We shall coordinatize generalized quadrangles with a regular spread by means of a Steiner system ( P , L ) , a set X and a certain nice map Δ : P × P → Sym ( X ) . We shall then show how this coordinatization method can be used to improve a result independently obtained by Kantor [W.M. Kantor, Note on span-symmetrical generalized quadrangles, Adv. Geom. 2 (2) (2002) 197–200] and Thas [K. Thas, Classification of span-symmetric generalized quadrangles of order s , Adv. Geom. 2 (2) (2002) 189–196] stating that a generalized quadrangle of order s ≥ 2 is isomorphic to W ( s ) if it has a hyperbolic line all of whose points are centres of symmetry. We shall show that if a generalized quadrangle Q of order s ≥ 2 has a hyperbolic line containing only regular points, then all these points are also centres of symmetry. Combining this with the above-mentioned result independently obtained by Kantor and Thas, we then obtain that Q is isomorphic to the symplectic quadrangle W ( s ) .

Corrigenda to "Polarities of Symplectic Quadrangles''

Corrigenda to "Polarities of Symplectic Quadrangles''

Polarities of Symplectic Quadrangles

We give a simple proof of the known fact that the symplectic quadrangle is self-dual if and only if the ground field is perfect of characteristic~2, and that a polarity exists exactly if there is a root of the Frobenius automorphism. Moreover, we determine all polarities, characterize the conjugacy classes of polarities, and use the results to give a simple proof that the centralizer of any polarity acts two-transitively on the ovoid of absolute points. The proofs use elementary calculations in solvable subgroups of the symplectic group.

Compact Three-Dimensional Elation Quadrangles

We show that each compact three-dimensional EGQ is isomorphic either to the real symplectic quadrangle, or to a translation quadrangle of Tits type. In particular, the elation group is one of the two classical elation groups, and thus a simply connected nilpotent Lie group.

Compact Generalised Quadrangles with Parameter 1 and Large Group of Automorphisms

If the group of automorphisms of a compact generalised quadrangle with parameter 1 has dimension at least 6, the quadrangle is the real symplectic quadrangle or its dual. There are nonclassical compact generalised quadrangles with parameter 1 whose group of automorphism has dimension 5.

Simple groups of finite morley rank and Tits buildings

Theorem A:If ℬ is an infinite Moufang polygon of finite Morley rank, then ℬ is either the projective plane, the symplectic quadrangle, or the split Cayley hexagon over some algebraically closed field. In particular, ℬ is an algebraic polygon.

Compact Ovoids in Quadrangles I: Geometric Constructions

This paper is about ovoids in infinite generalized quadrangles. Using the axiom of choice, Cameron showed that infinite quadrangles contain many ovoids. Therefore, we consider mainly closed ovoids in compact quadrangles. After deriving some basic properties of compact ovoids, we consider ovoids which arise from full imbeddings. This leads to restrictions for the topological parameters (m,m′). For example, if there is a regular pair of lines or a full closed subquadrangle, then m≤m′. The existence of full subquadrangles implies the nonexistence of ideal subquadrangles, so finite-dimensional quadrangles are either point-minimal or line-minimal. Another result is that (up to duality) such a quadrangle is spanned by the set of points on an ordinary quadrangle. This is useful for studying orbits of automorphism groups. Finally we prove general nonexistence results for ovoids in quadrangles with low-dimensional line pencils. As one consequence we show that the symplectic quadrangle over an algebraically closed field of characteristic 0 has no Zariski-closed ovoids or spreads.

Groups of projectivities of generalized quadrangles.

We characterize some classical quadrangles by means of properties of their groups of projectivities. In particular, we characterize all finite classical quadrangles with regular lines, and all symplectic quadrangles over quadratically closed fields.

A Combinatorial Characterization of Some Finite Classical Generalized Hexagons

We characterize the Moufang hexagons in characteristic 2 by requiring that certain sets of points (defined by distances from other objects) are nonempty. Together with a known result for quadrangles, we obtain a common combinatorial characterization of the symplectic quadrangles over an arbitrary finite field and the classical hexagons over a field of characteristic 2 amongst all finite generalized polygons.

A Geometric characterization of the perfect Suzuki-Tits ovoids

We characterize in a geometrical way those Suzuki-Tits ovoids which are defined over a perfect fieidK (or equivalently living inside a self polar symplectic quadrangle). We simplify our axioms in the particular cases that (1) the associated Suzuki group has exactly two orbits in the set of lines ofPG(3,K), and (2) the ovoid is finite.

Algebraic Polygons

In this paper we prove the following: Over each algebraically closed fieldKof characteristic 0 there exist precisely three algebraic polygons (up to duality), namely the projective plane, the symplectic quadrangle, and the split Cayley hexagon overK(Theorem 3.3). As a corollary we prove that every algebraic Tits system overKis Moufang and obtain the following classification:THEOREM.Let(G,B,N,S)be an irreducible effective spherical Tits system of rank≥2.If G is a connected algebraic group over an algebraically closed field of characteristic0,and if B is closed in G,then G is simple and B is a standard Borel subgroup of G.

Slanted symplectic quadrangles

By ‘slanting’ symplectic quadrangles W(F) over fieldsF, we obtain very simple examples of non-classical generalized quadrangles. We determine the collineation groups of these slanted quadrangles and their groups of projectivities. No slanted quadrangle is a topological quadrangle.

Characterizing symplectic quadrangles by their derivations

Characterizing symplectic quadrangles by their derivations