Synchronizing permutation groups and graph endomorphisms

Permutation groups;Endomorphisms (Group theory)

Schaefer, Artur ; Ruškuc, Nik ; Ruškuc, Nik

University:University of St Andrews

Department:Mathematics & Statistics (School of)

关键词: Permutation groups;    Endomorphisms (Group theory);   

Others  :  https://research-repository.st-andrews.ac.uk/bitstream/handle/10023/9912/ArturSchaeferPhDThesis.pdf?sequence=2&isAllowed=y

来源: DR-NTU

【 摘 要 】

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 boundon synchronizing ranks of rank 3 groups, at first. Then, we determine the singular graphendomorphisms of the Hamming graph and related graphs, count Latin hypercuboids ofclass r, establish their relation to mixed MDS codes, investigate G-decompositions of(non)-synchronizing semigroups, and analyse the kernel graph construction used in thetheorem of Cameron and Kazanidis which identifies non-synchronizing transformationswith 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 acomplete 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 arecharacterised (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 MDScodes is unveiled (see Chapter 6). 4. The research on normalizing groups from [3] is extended to semigroups of theform , and decomposition properties of non-synchronizing semigroups are described which are then applied to semigroups induced by combinatorial tilingproblems (see Chapter 7). 5. At last, it is shown that all rank 3 graphs admitting singular endomorphisms arehulls and it is conjectured that a hull on n vertices has minimal generating set of atmost n generators (see Chapter 8).