1,466 research outputs found
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
Symmetric graphs with complete quotients
Let be a -symmetric graph with vertex set . We suppose that
admits a -partition , with parts of
size , and that the quotient graph induced on is a complete
graph of order . Then, for each pair of distinct suffices , the
graph induced on the union is bipartite with each vertex of
valency or (a constant). When , it was shown earlier how a
flag-transitive -design induced on a part can sometimes be
used to classify possible triples . Here we extend
these ideas to and prove that, if the group induced by on a part
is -transitive and the "blocks" of have size less than ,
then either (i) , or (ii) the triple is known
explicitly.Comment: The first version of this manuscript dates from 2000. It was uploaded
to the arXiv since several people wished to have a copy. This new version is
updated with a literature review up to 2017. It is submitted for publication
and is currently under review (September 2017
Unitary graphs and classification of a family of symmetric graphs with complete quotients
A finite graph is called -symmetric if is a group of
automorphisms of which is transitive on the set of ordered pairs of
adjacent vertices of . We study a family of symmetric graphs, called
the unitary graphs, whose vertices are flags of the Hermitian unital and whose
adjacency relations are determined by certain elements of the underlying finite
fields. Such graphs admit the unitary groups as groups of automorphisms, and
they play a significant role in the classification of a family of symmetric
graphs with complete quotients such that an associated incidence structure is a
doubly point-transitive linear space. We give this classification in the paper
and also investigate combinatorial properties of the unitary graphs
Vertex-imprimitive symmetric graphs with exactly one edge between any two distinct blocks
A graph is called -symmetric if it admits as a group of
automorphisms acting transitively on the set of ordered pairs of adjacent
vertices. We give a classification of -symmetric graphs with
admitting a nontrivial -invariant partition such
that there is exactly one edge of between any two distinct blocks of
. This is achieved by giving a classification of -point-transitive and -block-transitive designs together
with -orbits on the flag set of such that is transitive on and for
distinct , where is the
setwise stabilizer of in the stabilizer of in .
Along the way we determine all imprimitive blocks of on for every -transitive group on a set , where
.Comment: This is the final version which will appear in JCT(A). The previous
title of this paper was "symmetric spreads of complete graphs
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
Steiner systems for supported by a family of extended cyclic codes
In [C. Ding, An infinite family of Steiner systems from cyclic
codes, {\em J. Combin. Des.} 26 (2018), no.3, 126--144], Ding constructed a
family of Steiner systems for all from a
family of extended cyclic codes. The objective of this paper is to present a
family of Steiner systems for all supported
by a family of extended cyclic codes. The main result of this paper complements
the previous work of Ding, and the results in the two papers will show that
there exists a binary extended cyclic code that can support a Steiner system
for all even . This paper also determines the parameters
of other -designs supported by this family of extended cyclic codes
Flag-transitive -designs and groups
This paper considers flag-transitive - designs with
and . Let the automorphism group of a design
be a simple group . Depend on the fact that the setwise stabilizer
must be one of twelve kinds of subgroups, up to isomorphism we get the
following two results. (i) If , then except
or
undecided, is
a -, -, -, -, -,
-, - or - design with
, , , , ,
, or respectively.
(ii) If , , , , ( even) or
, where , then there is no such design
Proofs of Two Conjectures On the Dimensions of Binary Codes
Let and be the binary codes generated by the
column -null space of the incidence matrix of external points
versus passant lines and internal points versus secant lines with respect to a
conic in , respectively. We confirm the conjectures on the dimensions
of and using methods from both finite geometry
and modular representation theory.Comment: 29 page
Symmetric designs and projective special unitary groups of dimension at most five
In this article, we study symmetric designs admitting a
flag-transitive and point-primitive automorphism group whose socle is a
projective special unitary group of dimension at most five. We, in particular,
determine all such possible parameters and show that there
exist eight non-isomorphic of such designs for which and is , , or
Large dimensional classical groups and linear spaces
Suppose that a group has socle a simple large-rank classical group.
Suppose furthermore that acts transitively on the set of lines of a linear
space . We prove that, provided has dimension at least 25,
then acts transitively on the set of flags of and hence the
action is known. For particular families of classical groups our results hold
for dimension smaller than 25.
The group theoretic methods used to prove the result (described in Section 3)
are robust and general and are likely to have wider application in the study of
almost simple groups acting on finite linear spaces.Comment: 32 pages. Version 2 has a new format that includes less repetition.
It also proves a slightly stronger result; with the addition of our
"Concluding Remarks" section the result holds for dimension at least 2
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