A characterization of the generalized quadrangle Q(4,q), q ODD

In [3], [ 4 ] we introduced the concept of (0,2)-set in generalized quadrangles, in order to obtain characterizations for P(S,()) and T ₂ ^* (O). Using these sets we are now able to formulate a characterization for Q(4,q), q odd, by assuming local conditions in an antiregular point x of a generalized quadrangle of order s.     

A geometric proof of a theorem on antiregularity of generalized quadrangles

A geometric proof is given in terms of Laguerre geometry of the theorem of Bagchi, Brouwer and Wilbrink, which states that if a generalized quadrangle of order s > 1 has an antiregular point then all of its points are antiregular.     

The uniqueness of 1-systems of W 5(q) satisfying the BLT-property, with q odd

In the symplectic polar space W ₅(q) every 1-system which satisfies the BLT-property (and then q is odd) defines a generalized quadrangle (GQ) of order (q ²,q ³). In this paper, we show that this 1-system is unique, so that the only GQ arising in this way is isomorphic to the classical GQ H(4,q ²), q odd.     

Subquadrangle m‐regular systems on generalized quadrangles

We introduce the notion of subquadrangle regular system of a generalized quadrangle. A subquadrangle regular system of order m on a generalized quadrangle of order (s, t) is a set ? of embedded subquadrangles with the property that every point lies on exactly m subquadrangles of ?. If m is one half of the total number of subquadrangles on a point, we call ? a subquadrangle hemisystem. We construct two infinite families of symplectic subquadrangle hemisystems of the Hermitian surface ??(3, q²), q odd, and two infinite families of symplectic subquadrangle hemisystems of ??₃(q²), q even. Some sporadic examples of symplectic subquadrangle regular systems of ??(3, q²) are also presented. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:28‐41, 2010     

Regular graphs constructed from the classical generalized quadrangle Q(4, q)

Let c_k be the smallest number of vertices in a regular graph with valency k and girth 8. It is known that c_{k + 1}?2(1 + k + k² + k³) with equality if and only if there exists a finite generalized quadrangle of order k. No such quadrangle is known when k is not a prime power. In this case, small regular graphs of valency k + 1 and girth 8 can be constructed from known generalized quadrangles of order q>k by removing a part of its structure. We investigate the case when q = k + 1 is a prime power, and try to determine the smallest graph under consideration that can be constructed from a generalized quadrangle of order q. This problem appears to be much more difficult than expected. We have general bounds and improve these for the classical generalized quadrangle Q(4, q), q even. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:70‐83, 2010     

Tight sets and <Emphasis Type="Italic">m</Emphasis>-ovoids of generalised quadrangles

The concept of a tight set of points of a generalised quadrangle was introduced by S. E. Payne in 1987, and that of an m-ovoid of a generalised quadrangle was introduced by J. A. Thas in 1989, and we unify these two concepts by defining intriguing sets of points. We prove that every intriguing set of points in a generalised quadrangle is an m-ovoid or a tight set, and we state an intersection result concerning these objects. In the classical generalised quadrangles, we construct new m-ovoids and tight sets. In particular, we construct m-ovoids of W(3,q), q odd, for all even m; we construct (q+1)/2-ovoids of W(3,q) for q odd; and we give a lower bound on m for m-ovoids of H(4,q ²).     

Generalized quadrangles associated with G2(q)

If q ≡ 2 (mod 3), a generalized quadrangle with parameters q, q² is constructed from the generalized hexagon associated with the group G₂(q).     

A Characterization of the Complement of a Hyperbolic Quadric in PG (3, q), for q Odd

The incidence structure NQ⁺(3, q) has points the points not on a non-degenerate hyperbolic quadric Q⁺(3, q) in PG(3, q), and its lines are the lines of PG(3, q) not containing a point of Q⁺(3, q). It is easy to show that NQ⁺(3, q) is a partial linear space of order (q, q(q−1)/2). If q is odd, then moreover NQ⁺(3, q) satisfies the property that for each non-incident point line pair (x,L), there are either (q−1)/2 or (q+1)/2 points incident with L that are collinear with x. A partial linear space of order (s, t) satisfying this property is called a ((q−1)/2,(q+1)/2)-geometry. In this paper, we will prove the following characterization of NQ⁺(3,q). Let S be a ((q−1)/2,(q+1)/2)-geometry fully embedded in PG(n, q), for q odd and q>3. Then S = NQ⁺(3, q).     

Generalized quadrangles of order (q,q 2),q even,containingW(q) as a subquadrangle

In this paper we give a characterization of the generalized quadrangleQ(5,q),q even, in terms of ovoids of its subquadrangles.     

A spectrum result on maximal partial ovoids of the generalized quadrangle , even

This article presents a spectrum result on maximal partial ovoids of the generalized quadrangle Q(4,q), q even. We prove that for every integer k in an interval of, roughly, size [q²/10,9q²/10], there exists a maximal partial ovoid of size k on Q(4,q), q even. Since the generalized quadrangle W(q), q even, defined by a symplectic polarity of PG(3,q) is isomorphic to the generalized quadrangle Q(4,q), q even, the same result is obtained for maximal partial ovoids of W(q), q even. As equivalent results, the same spectrum result is obtained for minimal blocking sets with respect to planes of PG(3,q), q even, and for maximal partial 1-systems of lines on the Klein quadric Q⁺(5,q), q even.     

On a set of lines of PG(3, q) corresponding to a maximal cap contained in the Klein quadric of PG (5, q)

The problem is considered of constructing a maximal set of lines, with no three in a pencil, in the finite projective geometry PG(3, q) of three dimensions over GF(q). (A pencil is the set of q+1 lines in a plane and passing through a point.) It is found that an orbit of lines of a Singer cycle of PG(3, q) gives a set of size q ³ + q ² + q + 1 which is definitely maximal in the case of q odd. A (q ³ + q ² + q + 1)-cap contained in the hyperbolic (or Klein) quadric of PG(5, q) also comes from the construction. (A k-cap is a set of k points with no three in a line.) This is generalized to give direct constructions of caps in quadrics in PG(5, q). For q odd and greater than 3 these appear to be the largest caps known in PG(5, q). In particular it is shown how to construct directly a large cap contained in the Klein quadric, given an ovoid skew to an elliptic quadric of PG(3, q). Sometimes the cap is also contained in an elliptic quadric of PG(5, q) and this leads to a set of q ³ + q ² + q + 1 lines of PG(3,q ²) contained in the non-singular Hermitian surface such that no three lines pass through a point. These constructions can often be applied to real and complex spaces.     

A hemisystem of a nonclassical generalised quadrangle

The concept of a hemisystem of a generalised quadrangle has its roots in the work of B. Segre, and this term is used here to denote a set of points such that every line ℓ meets in half of the points of ℓ. If one takes the point-line geometry on the points of the hemisystem, then one obtains a partial quadrangle and hence a strongly regular point graph. The only previously known hemisystems of generalised quadrangles of order (q, q ²) were those of the elliptic quadric , q odd. We show in this paper that there exists a hemisystem of the Fisher–Thas–Walker–Kantor generalised quadrangle of order (5, 5²), which leads to a new partial quadrangle. Moreover, we can construct from our hemisystem the 3· A ₇-hemisystem of , first constructed by Cossidente and Penttila.      

Blocking Sets Of External Lines To A Conic In PG(2,q), q ODD

We determine all point-sets of minimum size in PG(2,q), q odd that meet every external line to a conic in PG(2,q). The proof uses a result on the linear system of polynomials vanishing at every internal point to the conic and a corollary to the classification theorem of all subgroups of PGL(2,q). * Research supported by the Italian Ministry MURST, Strutture geometriche, combinatoria e loro applicazioni and by the Hungarian-Italian Intergovernemental project “Algebraic and Geometric Structures”.     

The affine planeAG(2,q),q odd,has a unique one point extension

Summary LetJ be a finite inversive plane of odd orderq. If for at least one pointp ofJ the internal affine planeJ _p is Desarguesian, thenJ is Miquelian. Other formulation: the finite Desarguesian affine plane of odd orderq has a unique one point extension; this extension is the Miquelian inversive plane of orderq. It follows that there is a unique inversive plane of orderq, withq{3, 5, 7}.Oblatum 23-X-1992 & 24-I-1994     

A New Characterization of PSL(p,q)

Abstract

Order components of a finite group are introduced in Chen [Chen, G. Y. (1996c) On Thompson's conjecture. J. Algebra 185:184–193]. It was proved that PSL(3, q), where q is an odd prime power, is uniquely determined by its order components [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002a). A characterization of PSL(3, q) where q is an odd prime power. J. Pure Appl. Algebra 170(2–3): 243–254]. Also in Iranmanesh et al. [Iranmanesh, A., Alavi, S. H., Khosravi, B. (2002b). A characterization of PSL(3, q) where q = 2 ⁿ . Acta Math. Sinica, English Ser. 18(3):463–472] and [Iranmanesh, A., Alavi, S. H. (2002). A characterization of simple groups PSL(5, q). Bull. Austral. Math. Soc. 65:211–222] it was proved that PSL(3, q) for q = 2 ⁿ and PSL(5, q) are uniquely determined by their order components. In this paper we prove that PSL(p, q) can be uniquely determined by its order components, where p is an odd prime number. A main consequence of our results is the validity of Thompson's conjecture for the groups under consideration.     

A characterization of PGL(2, q), q odd

Following the lines of [10], we give a characterization of the group PGL(2, q), q odd, in terms of involutions.Work performed under the auspicies of G.N.S.A.C.A. of C.N.R. supported by the 40% grants of M.P.I.     

New infinite families of hyperovals on \mathcal H (3,q^2), q odd

Two new infinite families of hyperovals on the generalized quadrangle \(\mathcal H (3,q^2), q\) odd, are constructed.     

A Family of Non-quasiprimitive Graphs Constructed From PSU3（q） Where q Is Odd

Let Г be a simple connected graph and let G be a group of automorphisms of Г. Г is said to be （G, 2）-arc transitive if G is transitive on the 2-arcs of Г. It has been shown that there exists a family of non-quasiprimitive （PSU3（q）, 2）-arc transitive graphs where q = 2^3m with m an odd integer. In this paper we investigate the case where q is an odd prime power.     

Proper 2-Covers of PG(3,q), q Even

A k-cover of =PG(3q) is a set S of lines of such that every point is on exactly k lines of S. S is proper if it contains no spread. The existence of proper k-covers of is necessary for the existence of maximal partial packings of q ²+q+1–k spreads of . Here we give the first construction of proper 2-packings of PG(3,q) with q even; for q odd these have been constructed by Ebert.     

Blocking sets of nonsecant lines to a conic in PG(2,q), q odd

In a previous paper 1 , all point sets of minimum size in PG(2,q), blocking all external lines to a given irreducible conic , have been determined for every odd q. Here we obtain a similar classification for those point sets of minimum size, which meet every external and tangent line to . © 2004 Wiley Periodicals, Inc.