4,217 research outputs found
The random graph
Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique
(and highly symmetric) countably infinite random graph. This graph, and its
automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul
Erd\H{o}s
Groups with right-invariant multiorders
A Cayley object for a group G is a structure on which G acts regularly as a
group of automorphisms. The main theorem asserts that a necessary and
sufficient condition for the free abelian group G of rank m to have the generic
n-tuple of linear orders as a Cayley object is that m>n. The background to this
theorem is discussed. The proof uses Kronecker's Theorem on diophantine
approximation.Comment: 9 page
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
Most primitive groups are full automorphism groups of edge-transitive hypergraphs
We prove that, for a primitive permutation group G acting on a set of size n,
other than the alternating group, the probability that Aut(X,Y^G) = G for a
random subset Y of X, tends to 1 as n tends to infinity. So the property of the
title holds for all primitive groups except the alternating groups and finitely
many others. This answers a question of M. Klin. Moreover, we give an upper
bound n^{1/2+\epsilon} for the minimum size of the edges in such a hypergraph.
This is essentially best possible.Comment: To appear in special issue of Journal of Algebra in memory of Akos
Seres
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