35 research outputs found
Covers of generalized quadrangles
We solve a problem posed by Cardinali and Sastry (Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order. Proc. Indian Acad. Sci. Math. Sci. 126 (2016), 591-612) about factorization of 2-covers of finite classical generalized quadrangles (GQs). To that end, we develop a general theory of cover factorization for GQs, and in particular, we study the isomorphism problem for such covers and associated geometries. As a byproduct, we obtain new results about semi-partial geometries coming from theta-covers, and consider related problems
A question of Frohardt on -groups, and skew translation quadrangles of even order
We solve a fundamental question posed in Frohardt's 1988 paper [Fro] on
finite -groups with Kantor familes, by showing that finite groups with a
Kantor family having distinct members such that is a central subgroup of and the
quotient is abelian cannot exist if the center of has
exponent and the members of are elementary abelian. In a
similar way, we solve another old problem dating back to the 1970s by showing
that finite skew translation quadrangles of even order are always
translation generalized quadrangles.Comment: 10 pages; submitted (February 2018
Generalized quadrangles of orrder (s, s2), I
AbstractIn this paper generalized quadrangles of order (s, s2), s > 1, satisfying property (G) at a line, at a pair of points, or at a flag, are studied. Property (G) was introduced by S. E. Payne (Geom. Dedicata32 (1989), 93â118) and is weaker than 3-regularity (see S. E. Payne and J. A. Thas, âFinite Generalized Quadrangles,â Pitman, London, 1984). It was shown by Payne that each generalized quadrangle of order (s2, s), s > 1, arising from a flock of a quadratic cone, has property (G) at its point (â). In particular translation generalized quadrangles satisfying property (G) are considered here. As an application it is proved that the Roman generalized quadrangles of Payne contain at least s3 + s2 classical subquadrangles Q(4, s). Also, as a by-product, several classes of ovoids of Q(4, s), s odd, are obtained; one of these classes is new. The goal of Part II is the classification of all translation generalized quadrangles satisfying property (G) at some flag ((â), L)
Geometrical Constructions of Flock Generalized Quadrangles
AbstractWith any flock F of the quadratic cone K of PG(3, q) there corresponds a generalized quadrangle S(F) of order (q2, q). For q odd Knarr gave a pure geometrical construction of S(F) starting from F. Recently, Thas found a geometrical construction of S(F) which works for any q. Here we show how, for q odd, one can derive Knarr's construction from Thas' one. To that end we describe an interesting representation of the point-plane flags of PG(3, q), which can be generalized to any dimension and which can be useful for other purposes. Applying this representation onto Thas' model of S(F), another interesting model of S(F) on a hyperbolic cone in PG(6, q) is obtained. In a final section we show how subquadrangles and ovoids of S(F) can be obtained via the description in PG(6, q)
Geometrical Constructions of Flock Generalized Quadrangles
AbstractWith any flock F of the quadratic cone K of PG(3, q) there corresponds a generalized quadrangle S(F) of order (q2, q). For q odd Knarr gave a pure geometrical construction of S(F) starting from F. Recently, Thas found a geometrical construction of S(F) which works for any q. Here we show how, for q odd, one can derive Knarr's construction from Thas' one. To that end we describe an interesting representation of the point-plane flags of PG(3, q), which can be generalized to any dimension and which can be useful for other purposes. Applying this representation onto Thas' model of S(F), another interesting model of S(F) on a hyperbolic cone in PG(6, q) is obtained. In a final section we show how subquadrangles and ovoids of S(F) can be obtained via the description in PG(6, q)
Covers of generalized quadrangles
We solve a problem posed by Cardinali and Sastry (Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order. Proc. Indian Acad. Sci. Math. Sci. 126 (2016), 591-612) about factorization of 2-covers of finite classical generalized quadrangles (GQs). To that end, we develop a general theory of cover factorization for GQs, and in particular, we study the isomorphism problem for such covers and associated geometries. As a byproduct, we obtain new results about semi-partial geometries coming from theta-covers, and consider related problems
Central aspects of skew translation quadrangles, I
Except for the Hermitian buildings , up to a combination
of duality, translation duality or Payne integration, every known finite
building of type satisfies a set of general synthetic
properties, usually put together in the term "skew translation generalized
quadrangle" (STGQ). In this series of papers, we classify finite skew
translation generalized quadrangles. In the first installment of the series, as
corollaries of the machinery we develop in the present paper, (a) we obtain the
surprising result that any skew translation quadrangle of odd order is
a symplectic quadrangle; (b) we determine all skew translation quadrangles with
distinct elation groups (a problem posed by Payne in a less general setting);
(c) we develop a structure theory for root-elations of skew translation
quadrangles which will also be used in further parts, and which essentially
tells us that a very general class of skew translation quadrangles admits the
theoretical maximal number of root-elations for each member, and hence all
members are "central" (the main property needed to control STGQs, as which will
be shown throughout); (d) we solve the Main Parameter Conjecture for a class of
STGQs containing the class of the previous item, and which conjecturally
coincides with the class of all STGQs.Comment: 66 pages; submitted (December 2013