A construction of one-dimensional affine flag-transitive linear spaces

AbstractThe finite flag-transitive linear spaces which have an insoluble automorphism group were given a precise description in [Francis Buekenhout, Anne Delandtsheer, Jean Doyen, Peter B. Kleidman, Martin W. Liebeck, Jan Saxl, Linear spaces with flag-transitive automorphism groups, Geom. Dedicata 36 (1) (1990) 89–94], and their classification has recently been completed (see [Martin W. Liebeck, The classification of finite linear spaces with flag-transitive automorphism groups of affine type, J. Combin. Theory Ser. A 84 (2) (1998) 196–235] and [Jan Saxl, On finite linear spaces with almost simple flag-transitive automorphism groups, J. Combin. Theory Ser. A 100 (2) (2002) 322–348]). However, the remaining case where the automorphism group is a subgroup of one-dimensional affine transformations has not been classified and bears a variety of known examples. Here we give a construction of new one-dimensional affine flag-transitive linear spaces via the André/Bruck–Bose construction applied to transitive line-spreads of projective space

Block-Transitive Designs in Affine Spaces

This paper deals with block-transitive $t$-$(v,k,\lambda)$ designs in affine spaces for large $t$, with a focus on the important index $\lambda=1$ case. We prove that there are no non-trivial 5-$(v,k,1)$ designs admitting a block-transitive group of automorphisms that is of affine type. Moreover, we show that the corresponding non-existence result holds for 4-$(v,k,1)$ designs, except possibly when the group is one-dimensional affine. Our approach involves a consideration of the finite 2-homogeneous affine permutation groups.Comment: 10 pages; to appear in: "Designs, Codes and Cryptography

Pairwise transitive 2-designs

We classify the pairwise transitive 2-designs, that is, 2-designs such that a group of automorphisms is transitive on the following five sets of ordered pairs: point-pairs, incident point-block pairs, non-incident point-block pairs, intersecting block-pairs and non-intersecting block-pairs. These 2-designs fall into two classes: the symmetric ones and the quasisymmetric ones. The symmetric examples include the symmetric designs from projective geometry, the 11-point biplane, the Higman-Sims design, and designs of points and quadratic forms on symplectic spaces. The quasisymmetric examples arise from affine geometry and the point-line geometry of projective spaces, as well as several sporadic examples.Comment: 28 pages, updated after review proces

The classification of flag-transitive Steiner 3-designs

We solve the long-standing open problem of classifying all 3-(v,k,1) designs with a flag-transitive group of automorphisms (cf. A. Delandtsheer, Geom. Dedicata 41 (1992), p. 147; and in: "Handbook of Incidence Geometry", ed. by F. Buekenhout, Elsevier Science, Amsterdam, 1995, p. 273; but presumably dating back to 1965). Our result relies on the classification of the finite 2-transitive permutation groups.Comment: 27 pages; to appear in the journal "Advances in Geometry