Abstract
If G is a transitive permutation group on a set X, then G is a Jordan group if there is
a partition of X into non-empty subsets Y and Z with |Z| > 1, such that the pointwise
stabilizer in G of Y acts transitively on Z (plus other non-degeneracy conditions).
There is a classification theorem by Adeleke and Macpherson for the infinite primitive
Jordan permutation groups: such group preserves linear-like structures, or tree-like
structures, or Steiner systems or a ‘limit’ of Steiner systems, or a ‘limit’ of betweenness
relations or D-relations. In this thesis we build a structure M whose automorphism
group is an infinite oligomorphic primitive Jordan permutation group preserving a limit
of D-relations.
In Chapter 2 we build a class of finite structures, each of which is essentially a finite lower
semilinear order with vertices labelled by finite D-sets, with coherence conditions. These
are viewed as structures in a relational language with relations L,L',S,S',Q,R. We
describe possible one point extensions, and prove an amalgamation theorem. We obtain
by Fra¨ıss´e’s Theorem a Fra¨ıss´e limit M.
In Chapter 3, we describe in detail the structure M and its automorphism group. We show
that there is an associated dense lower semilinear order, again with vertices labelled by
(dense) D-sets, again with coherence conditions.
By a method of building an iterated wreath product described by Cameron which is based
on Hall’s wreath power, we build in Chapter 4 a group K < Aut(M) which is a Jordan
group with a pre-direction as its Jordan set. Then we find, by properties of Jordan sets,
that a pre-D-set is a Jordan set for Aut(M). Finally we prove that the Jordan group
G = Aut(M) preserves a limit of D-relations as a main result of this thesis.
