415 research outputs found
Almost simple groups with socle acting on Steiner quadruple systems
Let , {}, a prime power, be a projective linear
simple group. We classify all Steiner quadruple systems admitting a group
with N \leq G \leq \Aut(N). In particular, we show that cannot act as a
group of automorphisms on any Steiner quadruple system for .Comment: 5 pages; to appear in: "Journal of Combinatorial Theory, Series A
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
Perfect countably infinite Steiner triple systems
We use a free construction to prove the existence of perfect Steiner triple systems on a countably infinite point set. We use a specific countably infinite family of partial Steiner triple systems to start the construction, thus yielding 2ℵ0 non-isomorphic perfect systems
Primitive decompositions of Johnson graphs
A transitive decomposition of a graph is a partition of the edge set together
with a group of automorphisms which transitively permutes the parts. In this
paper we determine all transitive decompositions of the Johnson graphs such
that the group preserving the partition is arc-transitive and acts primitively
on the parts.Comment: 35 page
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
Two Generalizations of Homogeneity in Groups with Applications to Regular Semigroups
Let be a finite set such that and let . A group
G\leq \sym is said to be -homogeneous if for every ,
such that and , there exists such that .
(Clearly -homogeneity is -homogeneity in the usual sense.)
A group G\leq \sym is said to have the -universal transversal property
if given any set (with ) and any partition of
into blocks, there exists such that is a section for .
(That is, the orbit of each -subset of contains a section for each
-partition of .)
In this paper we classify the groups with the -universal transversal
property (with the exception of two classes of 2-homogeneous groups) and the
-homogeneous groups (for ). As a
corollary of the classification we prove that a -homogeneous group is
also -homogeneous, with two exceptions; and similarly, but with no
exceptions, groups having the -universal transversal property have the
-universal transversal property.
A corollary of all the previous results is a classification of the groups
that together with any rank transformation on generate a regular
semigroup (for ).
The paper ends with a number of challenges for experts in number theory,
group and/or semigroup theory, linear algebra and matrix theory.Comment: Includes changes suggested by the referee of the Transactions of the
AMS. We gratefully thank the referee for an outstanding report that was very
helpful. We also thank Peter M. Neumann for the enlightening conversations at
the early stages of this investigatio
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