Abstract

An M × N matrix is associated with each ordered k-arc in a finite projective space of order N (k = M + N + 2.) The matrix is a projective invariant for ordered k-arcs in the space. The set of these matrices is denoted by Ω. Elements of the symmetric group Sk act on ordered arcs by permuting points. This induces a definition of Sk as a group of operators on Ω, whose orbits correspond to projectively distinct unordered k-arcs. Application of theorems of Burnside and Cauchy leads to results concerning the number of orbits of k-arcs in PG(N, q) under projectivity and under collineation. A subset of Ω is defined which contains representatives of each orbit under Sk. The reduced set of “normal” matrices is used by a counting algorithm. The results of this paper are applied to counting the projectively distinct unordered k-arcs (all k) in PG(2, 11) and PG(2, 13).

Introduction

This paper is concerned with the problem of counting equivalence classes of ordered k-arcs, unordered k-arcs, and k-gons with respect to either projectivity or collineation. Taking into account the three types of arc and two equivalence relations, there are six distinct but related problems to be solved. The six problems can each be stated in terms of the orbits of ordered arcs under some group of operators. The solutions, themselves, fall into two categories: formulas for the number of orbits and algorithms for counting the orbits.

Abstract

Let K be the quadratic cone with vertex υ in PG(3,q). A partition of K − {υ} into q disjoint irreducible conies is called a flock of K. A set of disjoint irreducible conies of K is called a partial flock of K. With each flock of K there corresponds a translation plane of order q2 and also a generalized quadrangle of order (q2, q). In this paper a partial flock F of size 11 is constructed for any q ≡ − 1 mod 12. For q = 11 F is a flock not isomorphic to any previously known flock. The subgroup G of PGL(4, q) fixing K and F is isomorphic to C2 × S3, hence has order 12. Finally, with the aid of a computer, it was shown by De Clerck and Herssens [3] that for q = 11 the cone K has exactly 4 mutually non-isomorphic flocks.

Introduction

Flocks of quadratic cones

Let K be the quadratic cone with vertex υ in PG(3, q). A partition of K − {υ} into q disjoint irreducible conies is called a flock of K. The flock F is called linear if the q planes of the conies of F all contain a common line L.

A set of disjoint irreducible conies of the cone K is called a partial flock of K.

Abstract

We study block-transitive t-(v, k, λ) designs for large t. We show that there are no nontrivial block-transitive 8-designs, and no nontrivial flag-transitive 7-designs. There are no known nontrivial block-transitive 6-designs; we show that the automorphism group of such a design, or of a flag-transitive 5-design with more than 24 points, must be either an affine group over GF(2) or a 2-dimensional projective linear group. We begin the investigation of these two cases, and construct a flag-transitive 5-(256, 24, λ) design for a suitable value of λ.

Introduction

A t-(v, k, λ) design is a pair D = (X, B), where X is a set of v points, B a set of k-element subsets of X called blocks, such that any t points are contained in exactly λ blocks, for some t ≤ k and λ > 0. Such a design (X, B) is said to be trivial if B consists of all the k-element subsets of X. A flag in a design D is an incident point-block pair. A subgroup G of the automorphism group of D is said to be block-transitive if G is transitive on B ; D is block-transitive if Aut(D) is. Point- and flag-transitivity are defined similarly. For information about t-designs, see Hughes and Piper [11].

In this paper we consider nontrivial block-transitive t-designs with t large. We use a result of Ray-Chaudhuri and Wilson [16] together with the finite simple group classification to show in Section 2 that t ≤ 7.

Abstract

An alternative construction for the dual G2(q)-hexagon is given for q odd and different from 3n.

Introduction

In, W.M. Kantor has constructed the generalized quadrangle associated with the Fisher-Thas-Walker flock as a group coset geometry starting from the dual G2(q)-hexagon. Analyzing Kantor's construction, the following question arises in a natural way: is it possible to define new points and new lines in a generalized quadrangle Q associated with a flock of the quadratic cone, in such a way that the new point-line geometry H is a generalized hexagon?

For q odd, we prove that the only possibility is that Q is the Kantor generalized quadrangle constructed in and H is the dual G2(q)-hexagon. If q ≠ 3n, using a twisted cubic of PG(3, q) we obtain an alternative construction of the dual G2(q)-hexagon similar to the construction of a generalized quadrangle using a BLT-set. For q even, we are able to prove a strong connection between the existence of H and the (q+1)-arcs of PG(3, q) but the answer is not complete due to difficulties of the same type that arise when studying BLT-sets in even characteristic.

We would like to express our thanks to S. E. Payne, J. A. Thas and H. Van Maldeghem for critical remarks on earlier versions of this paper, and to W. M. Kantor for useful discussions during his visit in Rome. In particular, Theorem 2.1 generalizes a result of W. M. Kantor (private communication).

Abstract

Let πl∞ be a semifield plane of order qN, with middle nucleus GF(q). Relative to any fixed natural autotopism triangle, every fixed affine point I (not on the triangle) determines up to isomorphism a unique coordinatising semifield DI. I is called a central unit if DI is commutative. We determine the geometric distribution of the central units of πl∞ and hence show that the plane has precisely (qN – l)(q – 1) central units.

Introduction

Let πl∞ be a semifield plane with an autotopism triangle OXY: where XY = l∞ is the translation axis, and OY is a shears axis, with O ∈ πl∞. Now each choice of a “unit point” I, off the chosen autotopism triangle (assumed fixed from now on) determines uniquely up to isomorphism a semifield DI that coordinatises πl∞. We shall call I a central unit relative to the chosen frame if DI is a commutative semifield. By a criterion of Ganley [2] [theorem 3] the finite semifield planes admitting central units are precisely the finite translation planes that admit orthogonal polarities. However, no geometric characterisation of the set of central units in a given semifield plane has ever been recorded. The purpose of this note is to provide a geometric description of the distribution of the central units of a given commutative semifield plane.

This book contains articles based on talks at the Second International Conference on Finite Geometry and Combinatorics, which took place from 31 May to 6 June 1992, at the Conference center De Ceder in Astene–Deinze, Belgium. There were 76 participants and 52 talks.

There are 35 articles in these proceedings.

The editors, who were also the conference organisers, are grateful for the financial support of the National Fund for Scientific Research (NFWO) of Belgium and of the University of Ghent.

Above all we are grateful to Zita Oost, the conference secretary, and to Ruth Lauwaert, who helped a great deal with the production of the copy.

Abstract

A generalized Fischer space is a partial linear space in which any two intersecting lines generate a subspace isomorphic to an affine plane or the dual of an affine plane. We give a classification of all finite and infinite generalized Fischer spaces under some nondegeneracy conditions.

Introduction

Let Π = (P, L) be a partial linear space, that is a set of points P together with a set L of subsets of P of cardinality at least 2 called lines, such that each pair of points is in at most one line. A subset X of P is called a subspace of Π if it has the property that any line meeting X in at least two points is contained in X. A subspace X together with the lines contained in it is a partial linear space. Subspaces are usualy identified with these partial linear spaces. As the intersection of any collection of subspaces is again a subspace, we can define for each subset X of P the subspace generated by X to be the smallest subspace containing X. This subspace will be denoted by 〈X〉. A plane is a subspace generated by two intersecting lines.

In [3], [6] partial linear spaces are considered in which all planes are either isomorphic to an affine plane or the dual of an affine plane. Such spaces are called generalized Fischer spaces.