A Census Of Highly Symmetric Combinatorial Designs

As a consequence of the classification of the finite simple groups, it has been possible in recent years to characterize Steiner t-designs, that is t-(v,k,1) designs, mainly for t = 2, admitting groups of automorphisms with sufficiently strong symmetry properties. However, despite the finite simple group classification, for Steiner t-designs with t > 2 most of these characterizations have remained longstanding challenging problems. Especially, the determination of all flag-transitive Steiner t-designs with 2 < t < 7 is of particular interest and has been open for about 40 years (cf. [11, p. 147] and [12, p. 273], but presumably dating back to 1965). The present paper continues the author's work [20, 21, 22] of classifying all flag-transitive Steiner 3-designs and 4-designs. We give a complete classification of all flag-transitive Steiner 5-designs and prove furthermore that there are no non-trivial flag-transitive Steiner 6-designs. Both results rely on the classification of the finite 3-homogeneous permutation groups. Moreover, we survey some of the most general results on highly symmetric Steiner t-designs.Comment: 26 pages; to appear in: "Journal of Algebraic Combinatorics

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

Steiner t-designs for large t

One of the most central and long-standing open questions in combinatorial design theory concerns the existence of Steiner t-designs for large values of t. Although in his classical 1987 paper, L. Teirlinck has shown that non-trivial t-designs exist for all values of t, no non-trivial Steiner t-design with t > 5 has been constructed until now. Understandingly, the case t = 6 has received considerable attention. There has been recent progress concerning the existence of highly symmetric Steiner 6-designs: It is shown in [M. Huber, J. Algebr. Comb. 26 (2007), pp. 453-476] that no non-trivial flag-transitive Steiner 6-design can exist. In this paper, we announce that essentially also no block-transitive Steiner 6-design can exist.Comment: 9 pages; to appear in: Mathematical Methods in Computer Science 2008, ed. by J.Calmet, W.Geiselmann, J.Mueller-Quade, Springer Lecture Notes in Computer Scienc

A common characterization of finite projective spaces and affine planes

Let S be a finite linear space for which there is a non-negative integer s such that for any two disjoint lines L, L' of S and any point p outside L and L' there are exactly s lines through p intersecting the two lines L and L'. We prove that one of the following possibilities occurs: S is a generalized projective space, and if the dimension of S is at least 4, then any line of S has exactly two points. S is an affine plane, an affine plane with one improper point, or a punctured projective plane. S is the Fano-quasi -plane. © 1981, Academic Press Inc. (London) Limited. All rights reserved.Math. Reviews 82j:51015SCOPUS: ar.jinfo:eu-repo/semantics/publishe

On a theorem of Wielandt for finite primitive permutation groups

Let G be a finite primitive permutation group with a non-trivial, non-regular normal subgroup N, and let γ be an orbit of a point stabilizer Nα. Then each composition factor S of N α occurs as a section of the permutation group induced by Nα on F. The case N = G is a theorem of Wielandt. The general result and some of its corollaries are useful for studying automorphism groups of combinatorial structures.(Math.Reviews 2004 j:20004)SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Finite line-transitive linear spaces: parameters and normal point-partitions

Until the 1990's the only known finite linear spaces admitting line-transitive, pointimprimitive groups of automorphisms were Desarguesian projective planes and two linear spaces with 91 points and line size 6. In 1992 a new family of 467 such spaces was constructed, all having 729 points and line size 8. These were shown to be the only linear spaces attaining an upper bound of Delandtsheer and Doyen on the number of points. Projective planes, and the linear spaces just mentioned on 91 or 729 points, are the only known examples of such spaces, and in all cases the line-transitive group has a non-trivial normal subgroup intransitive on points. The orbits of this normal subgroup form a partition of the point set called a normal point-partition. We give a systematic analysis of finite line-transitive linear spaces with normal point-partitions. As well as the usual parameters of linear spaces there are extra parameters connected with the normal point-partition that affect the structure of the linear space. Using this analysis we characterise the line-transitive linear spaces for which the values of various of these parameters are small. In particular we obtain a classification of all imprimitive linetransitive linear spaces that ‘nearly attain’ the Delandtsheer-Doyen upper bound. © 2003, by Walter de Gruyter GmbH & Co. KG. All rights reserved.(Math.Reviews 2004k :20002)SCOPUS: ar.jinfo:eu-repo/semantics/publishe

Finite linear spaces with flag-transitive groups

This paper is devoted to the study of finite linear spaces with a flag-transitive automorphism group. We survey known facts and introduce new results whose aim is to prepare a classification of such spaces and groups. In Section 1, we discuss various transitivity properties in finite linear spaces, and the relations between these properties. In Section 2, we give a list of examples of flag-transitive finite linear spaces, and the corresponding groups. In Section 3, we present some useful consequences of flag-transitivity. In Sections 4 and 5, we use the O'Nan-Scott theorem on primitive permutation groups to prove our main result: any group acting flag-transitively on a finite linear space is either of affine type or of simple type. In Section 6, we prove as a corollary that, with only one exception, any group acting transitively on the lines of a finite affine space must contain the translation group. © 1988.Math. Reviews 89k:20007SCOPUS: ar.jinfo:eu-repo/semantics/publishe