John Bamberg

Tight sets and m-ovoids of finite polar spaces

An intriguing set of points of a generalised quadrangle was introduced in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] as a unification of the pre-existing notions of tight set and m-ovoid. It was shown in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] that every intriguing set of points in a finite generalised quadrangle is a tight set or an m-ovoid (for some m). Moreover, it was shown that an m-ovoid and an i-tight set of a common generalised quadrangle intersect in mi points. These results yielded new proofs of old results, and in this paper, we study the natural analogue of intriguing sets in finite polar spaces of higher rank. In particular, we use the techniques developed in this paper to give an alternative proof of a result of Thas [J.A. Thas, Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10 (1-4) (1981) 135-143] that there are no ovoids of H(2r,q^2), Q^-(2r+1,q), and W(2r-1,q) for r>2. We also strengthen a result of Drudge on the non-existence of tight sets in W(2r-1,q), H(2r+1,q^2), and Q^+(2r+1,q), and we give a new proof of a result of De Winter, Luyckx, and Thas [S. De Winter, J.A. Thas, SPG-reguli satisfying the polar property and a new semipartial geometry, Des. Codes Cryptogr. 32 (1-3) (2004) 153-166; D. Luyckx, m-Systems of finite classical polar spaces, PhD thesis, The University of Ghent, 2002] that an m-system of W(4m+3,q) or Q^-(4m+3,q) is a pseudo-ovoid of the ambient projective space.

Tight sets and m -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,q2).

Finite permutation groups with a transitive minimal normal subgroup

A finite permutation group is said to be innately transitive if it contains a transitive minimal normal subgroup. In this paper, we give a characterisation and structure theorem for the finite innately transitive groups, as well as describing those innately transitive groups which preserve a product decomposition. The class of innately transitive groups contains all primitive and quasiprimitive groups.

Every flock generalized quadrangle has a hemisystem

We prove that every flock generalised quadrangle contains a hemisystem, and we provide a construction method which unifies our results with the examples of Cossidente and Penttila in the classical case.

Overgroups of cyclic Sylow subgroups of linear groups

We use a theorem of Guralnick, Penttila, Praeger, and Saxl to classify the subgroups of the general linear group (of a finite dimensional vector space over a finite field) which are overgroups of a cyclic Sylow subgroup. In particular, our results provide the starting point for the classification of transitive m-systems; which include the transitive ovoids and spreads of finite polar spaces. We also use our results to prove a conjecture of Cameron and Liebler on irreducible collineation groups having equally many orbits on points and on lines.

The Crystallographic Restriction, Permutations, and Goldbach's Conjecture

1. INTRODUCTION. The object of this paper is to make an observation connecting Goldbachs conjecture, the crystallographic restriction, and the orders of the elements of the symmetric group. First recall that for an element g of a group G the order Ord(g) of g is defined to be the smallest natural number such that g

Symplectic Spreads

We construct an infinite family of symplectic spreads in spaces of odd rank and characteristic.

Non-free points for groups generated by a pair of 2 × 2 matrices

A point λ in the complex plane is said to be free if the group generated by the matrices formula here is free. The paper gives an infinite family of polynomials whose roots are the non-free points. The main idea in the paper is to employ a symmetry relation.

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