Translations and axial collineations of affine designs

Abstract

Some types of affine (or projective) planes can be constructed from groups satisfying certain conditions such that the group then acts as a collineation group of the plane constructed from it. For example, a translation plane (an affine plane with a point transitive translation group) can be constructed from a group with a special type of subgroup partition such that the group then acts as the translation group of the plane. This approach is due to Andr6 (see [2] for details and references). Schulz [10] considered the more general situation of affine 2-designs with point transitive translation groups (finite affine planes are the affine 2-designs with 4=1). As a further example (see [5-]), a sharply 2-transitive permutation group of degree m can be represented as a collineation group of a translation plane of order m also admitting certain axial collineations, namely a plane coordinatized by a nearfield. In this paper we shall consider affine 2-designs admitting a collineation group containing translations (in one direction) and axial collineations such that the group acts sharply 2-transitive on the blocks of some parallel class. In the case of affine planes this situation arises in planes coordinatized by nearfields. In the first section we establish some results on collineations of affine 2-designs and introduce the concept of a quasicentre. Then we show that if an affine 2-design admits a collineation as described above, then the design decomposes into two smaller designs. Finally we show that this decomposition can be reversed by constructing an affine 2-design with this property from two smaller designs and a sharply 2-transitive permutation group.