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Title: Synchronizing permutation groups and graph endomorphisms
Author: Schaefer, Artur
ISNI:       0000 0004 5991 2805
Awarding Body: University of St Andrews
Current Institution: University of St Andrews
Date of Award: 2016
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The current thesis is focused on synchronizing permutation groups and on graph endo- morphisms. Applying the implicit classification of rank 3 groups, we provide a bound on synchronizing ranks of rank 3 groups, at first. Then, we determine the singular graph endomorphisms of the Hamming graph and related graphs, count Latin hypercuboids of class r, establish their relation to mixed MDS codes, investigate G-decompositions of (non)-synchronizing semigroups, and analyse the kernel graph construction used in the theorem of Cameron and Kazanidis which identifies non-synchronizing transformations with graph endomorphisms [20]. The contribution lies in the following points: 1. A bound on synchronizing ranks of groups of permutation rank 3 is given, and a complete list of small non-synchronizing groups of permutation rank 3 is provided (see Chapter 3). 2. The singular endomorphisms of the Hamming graph and some related graphs are characterised (see Chapter 5). 3. A theorem on the extension of partial Latin hypercuboids is given, Latin hyper- cuboids for small values are counted, and their correspondence to mixed MDS codes is unveiled (see Chapter 6). 4. The research on normalizing groups from [3] is extended to semigroups of the form < G, T >, and decomposition properties of non-synchronizing semigroups are described which are then applied to semigroups induced by combinatorial tiling problems (see Chapter 7). 5. At last, it is shown that all rank 3 graphs admitting singular endomorphisms are hulls and it is conjectured that a hull on n vertices has minimal generating set of at most n generators (see Chapter 8).
Supervisor: Ruškuc, Nik ; Cameron, Peter Jephson Sponsor: Engineering and Physical Sciences Research Council (EPSRC) ; University of St Andrews
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA175.S3 ; Permutation groups ; Endomorphisms (Group theory)