6 research outputs found
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
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
On the existence of block-transitive combinatorial designs
Block-transitive Steiner -designs form a central part of the study of
highly symmetric combinatorial configurations at the interface of several
disciplines, including group theory, geometry, combinatorics, coding and
information theory, and cryptography. The main result of the paper settles an
important open question: There exist no non-trivial examples with (or
larger). The proof is based on the classification of the finite 3-homogeneous
permutation groups, itself relying on the finite simple group classification.Comment: 9 pages; to appear in "Discrete Mathematics and Theoretical Computer
Science (DMTCS)
On the Cameron-Praeger Conjecture
This paper takes a significant step towards confirming a long-standing and
far-reaching conjecture of Peter J. Cameron and Cheryl E. Praeger. They
conjectured in 1993 that there are no non-trivial block-transitive 6-designs.
We prove that the Cameron-Praeger conjecture is true for the important case of
non-trivial Steiner 6-designs, i.e. for 6- designs with
, except possibly when the group is P\GammaL(2,p^e) with or
3, and is an odd prime power.Comment: 11 pages; to appear in: "Journal of Combinatorial Theory, Series A
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
The Classification of Flag-transitive Steiner 4-Designs
Among the properties of homogeneity of incidence structures flag-transitivity
obviously is a particularly important and natural one. Consequently, in the
last decades also flag-transitive Steiner tdesigns (i.e. flag-transitive
t-(v,k,1) designs) have been investigated, whereas only by the use of the
classification of the finite simple groups has it been possible in recent years
to essentially characterize all flag-transitive Steiner 2-designs. However,
despite the finite simple group classification, for Steiner t-designs with
parameters t > 2 such characterizations have remained challenging open problems
for about 40 years (cf. [11, p. 147] and [12, p. 273], but presumably dating
back to around 1965). The object of the present paper is to give a complete
classification of all flag-transitive Steiner 4-designs. Our result relies on
the classification of the finite doubly transitive permutation groups and is a
continuation of the author's work [20, 21] on the classification of all
flag-transitive Steiner 3-designs.Comment: 26 pages; to appear in: "Journal of Algebraic Combinatorics