731 research outputs found
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
Basic and degenerate pregeometries
We study pairs , where is a 'Buekenhout-Tits'
pregeometry with all rank 2 truncations connected, and is transitive on the set of elements of each type. The family of such
pairs is closed under forming quotients with respect to -invariant
type-refining partitions of the element set of . We identify the
'basic' pairs (those that admit no non-degenerate quotients), and show, by
studying quotients and direct decompositions, that the study of basic
pregeometries reduces to examining those where the group is faithful and
primitive on the set of elements of each type. We also study the special case
of normal quotients, where we take quotients with respect to the orbits of a
normal subgroup of . There is a similar reduction for normal-basic
pregeometries to those where is faithful and quasiprimitive on the set of
elements of each type
Primitive Groups Synchronize Non-uniform Maps of Extreme Ranks
Let be a set of cardinality , a permutation group on
, and a map which is not a permutation. We say that
synchronizes if the semigroup contains a constant
map.
The first author has conjectured that a primitive group synchronizes any map
whose kernel is non-uniform. Rystsov proved one instance of this conjecture,
namely, degree primitive groups synchronize maps of rank (thus, maps
with kernel type ). We prove some extensions of Rystsov's
result, including this: a primitive group synchronizes every map whose kernel
type is . Incidentally this result provides a new
characterization of imprimitive groups. We also prove that the conjecture above
holds for maps of extreme ranks, that is, ranks 3, 4 and .
These proofs use a graph-theoretic technique due to the second author: a
transformation semigroup fails to contain a constant map if and only if it is
contained in the endomorphism semigroup of a non-null (simple undircted) graph.
The paper finishes with a number of open problems, whose solutions will
certainly require very delicate graph theoretical considerations.Comment: Includes changes suggested by the referee of the Journal of
Combinatorial Theory, Series B - Elsevier. We are very grateful to the
referee for the detailed, helpful and careful repor
Brauer relations in finite groups
If G is a non-cyclic finite group, non-isomorphic G-sets X, Y may give rise
to isomorphic permutation representations C[X] and C[Y]. Equivalently, the map
from the Burnside ring to the representation ring of G has a kernel. Its
elements are called Brauer relations, and the purpose of this paper is to
classify them in all finite groups, extending the Tornehave-Bouc classification
in the case of p-groups.Comment: 39 pages; final versio
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