Almazaydeh, Asma Ibrahim A. (2019) Infinite Jordan Permutation Groups. PhD thesis, University of Leeds.
Abstract
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.
Metadata
Supervisors: | Macpherson, H. Dugald |
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Keywords: | Jordan groups, Permutation groups, Fra¨ıss´e’s Theorem |
Awarding institution: | University of Leeds |
Academic Units: | The University of Leeds > Faculty of Maths and Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Identification Number/EthosID: | uk.bl.ethos.778708 |
Depositing User: | Mrs Asma Almazaydeh |
Date Deposited: | 25 Jun 2019 11:56 |
Last Modified: | 11 Jul 2020 09:53 |
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