178 research outputs found
On flag-transitive automorphism groups of symmetric designs
In this article, we study flag-transitive automorphism groups of non-trivial
symmetric designs, where divides and . We show that such an automorphism group is either point-primitive
of affine or almost simple type, or point-imprimitive with parameters
and , for some positive
integer . We also provide some examples in both possibilities
On Pseudocyclic Association Schemes
The notion of pseudocyclic association scheme is generalized to the
non-commutative case. It is proved that any pseudocyclic scheme the rank of
which is much more than the valency is the scheme of a Frobenius group and is
uniquely determined up to isomorphism by its intersection number array. An
immediate corollary of this result is that any scheme of prime degree, valency
and rank at least is schurian.Comment: 23 pages, 7 figure
Between primitive and -transitive: Synchronization and its friends
An automaton is said to be synchronizing if there is a word in the
transitions which sends all states of the automaton to a single state. Research
on this topic has been driven by the \v{C}ern\'y conjecture, one of the oldest
and most famous problems in automata theory, according to which a synchronizing
-state automaton has a reset word of length at most . The
transitions of an automaton generate a transformation monoid on the set of
states, and so an automaton can be regarded as a transformation monoid with a
prescribed set of generators. In this setting, an automaton is synchronizing if
the transitions generate a constant map. A permutation group on a set
is said to synchronize a map if the monoid
generated by and is synchronizing in the above sense; we say is
synchronizing if it synchronizes every non-permutation.
The classes of synchronizing groups and friends form an hierarchy of natural
and elegant classes of groups lying strictly between the classes of primitive
and -homogeneous groups. These classes have been floating around for some
years and it is now time to provide a unified reference on them. The study of
all these classes has been prompted by the \v{C}ern\'y conjecture, but it is of
independent interest since it involves a rich mix of group theory,
combinatorics, graph endomorphisms, semigroup theory, finite geometry, and
representation theory, and has interesting computational aspects as well. So as
to make the paper self-contained, we have provided background material on these
topics. Our purpose here is to present results that show the connections
between the various areas of mathematics mentioned above, we include a new
result on the \v{C}ern\'y conjecture, some challenges to finite geometers, some
thoughts about infinite analogues, and a long list of open problems
Block-transitive automorphism groups on 3-designs with small block size
The paper is an investigation of the structure of block-transitive
automorphism groups of a 3-design with small block size. Let be a
block-transitive automorphism group of a nontrivial - design
with . We prove that if is point-primitive then
is of affine or almost simple type. If is point-imprimitive then
is a - design with , and .Comment: 13 page
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
Flag-transitive point-primitive non-symmetric 2-(v,k,2) designs with alternating socle
This paper studies flag-transitive point-primitive non-symmetric
-() designs. We prove that if is a non-trivial
non-symmetric - design admitting a flag-transitive point-primitive
automorphism group with for , then is
a - or - design.Comment: 17 pages, 3 figuer
Self-dual Codes over the Kleinian Four Group
We introduce self-dual codes over the Kleinian four group K = Z_2 x Z_2 for a
natural quadratic form on K^n and develop the theory.
Topics studied are: weight enumerators, mass formulas, classification up to
length 8, neighbourhood graphs, extremal codes, shadows, generalized t-designs,
lexicographic codes, the Hexacode and its odd and shorter cousin, automorphism
groups, marked codes.
Kleinian codes form a new and natural fourth step in a series of analogies
between binary codes, lattices and vertex operator algebras. This analogy will
be emphasized and explained in detail.Comment: 26 pages with 5 tables and 1 figure, LaTe
Finite -connected homogeneous graphs
A finite graph \G is said to be {\em -connected homogeneous}
if every isomorphism between any two isomorphic (connected) subgraphs of order
at most extends to an automorphism of the graph, where is a
group of automorphisms of the graph. In 1985, Cameron and Macpherson determined
all finite -homogeneous graphs. In this paper, we develop a method for
characterising -connected homogeneous graphs. It is shown that for a
finite -connected homogeneous graph \G=(V, E), either G_v^{\G(v)} is
--transitive or G_v^{\G(v)} is of rank and \G has girth , and
that the class of finite -connected homogeneous graphs is closed under
taking normal quotients. This leads us to study graphs where is
quasiprimitive on . We determine the possible quasiprimitive types for
in this case and give new constructions of examples for some possible types
On cyclotomic schemes over finite near-fields
We introduce a concept of cyclotomic association scheme C over a finite
near-field. It is proved that if C is nontrivial, then Aut(C)<AGL(V) where V is
the linear space associated with the near-field. In many cases we are able to
get more specific information about Aut(C)
Flag-transitive non-symmetric -designs with and exceptional groups of Lie type
This paper determined all pairs where is a
non-symmetric 2- design with and is the
almost simple flag-transitive automorphism group of with an
exceptional socle of Lie type. We prove that if
where is an exceptional group of Lie type, then must be the Ree group
or Suzuki group, and there are five classes of non-isomorphic designs
- …