The only known enumeration of Buekenhout unitals occurs in the Desarguesian plane PG(2, q2). In this paper we develop general techniques for enumerating the nonsingular Buekenhout unitals embedded in any two-dimensional translation plane, and apply these techniques to obtain such an enumeration in the regular nearfield planes, the odd-order Hall planes, and the flag-transitive affine planes. We...

In earlier work we gave a characterisation of pregeometries which are ‘basic’ (that is, admit no ‘non-degenerate’ quotients) relative to two different kinds of quotient operation, namely taking imprimitive quotients and normal quotients. Each basic geometry was shown to involve a faithful group action, which is primitive or quasiprimitive, respectively, on the set of elements of each type. For ...

We offer a sufficient condition for a closed subset in an association scheme to be maximal. The result generalizes naturally the well-known (group-theoretical) fact that the one-point-stabilizer of a flag transitive automorphism group of a 2-design with *=1 must be a maximal subgroup. In finite group theory, there exist a lot of (important) sufficient conditions for a subgroup to be maximal. In...

In several areas of mathematics, homogeneous spaces are fundamental objects as they often serve as models for more general objects: Various examples from differential geometry (Riemannian symmetric spaces, principal bundles) and topology (geometric 3-manifolds), to algebraic and complex geometry (uniformization theorems, flag manifolds) etc. underline the importance of spaces, furnished with a ...

Let (P, L, *) be a near polygon having 8 + 1 points per line, s > 1, and suppose k is a field. Let Vk be the k-vector space with basis {vp | p € P}. Then the subspace generated by the vectors v1 = Lp*l vp, where l 6 L, has codimension at least 2 in Vk. This observation is used in two ways. First we derive the existence of certain diagram geometries with flag transitive automorphism group, and s...

Points will always be denoted by small latin letters, lines by capitals (unlike Bonisoli’s notations). By (a, b), with a, b ∈ N, we denote the greatest common divisor of a and b. Suppose π is a projective plane of order n, and suppose (p, L) is a point-line pair. Then a collineation θ of π is a (p, L)-collineation if θ fixes any point on L and every line through p. If (p, L) is a flag, then θ i...

In this article, we study symmetric designs D with parameters ( v , k λ ) admitting a flag-transitive and point-primitive automorphism group G whose socle X is finite simple exceptional of Lie type. conclusion, the design q 6 − 1 / 5 4 associated to generalised hexagon H or its dual = 2 point-stabiliser ˆ [ ] : L orthogonal 351 126 45 378 117 36 respectively for ϵ + 3 S . Our analysis depends h...