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dc.contributor.advisorRuškuc, Nik
dc.contributor.advisorCameron, Peter J. (Peter Jephson)
dc.contributor.authorSchaefer, Artur
dc.coverage.spatial248 p.en_US
dc.description.abstractThe 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).en_US
dc.publisherUniversity of St Andrews
dc.subject.lcshPermutation groupsen
dc.subject.lcshEndomorphisms (Group theory)en
dc.titleSynchronizing permutation groups and graph endomorphismsen_US
dc.contributor.sponsorEngineering and Physical Sciences Research Council (EPSRC)en_US
dc.contributor.sponsorUniversity of St Andrews. School of Mathematics and Statisticsen_US
dc.type.qualificationnamePhD Doctor of Philosophyen_US
dc.publisher.institutionThe University of St Andrewsen_US

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