Ph.D. Theses
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Browsing Ph.D. Theses by Author "Coşkun, Olcay."
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Item Composition factors of the functor of the complex characters(Thesis (Ph.D.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2018., 2018.) Arslan, Mehmet.; Coşkun, Olcay.When we consider CRC as a map sending any finite group G to the complex vector space CRC(G) of complex valued class functions on G, it becomes an A-fibered biset functor for any group A ≤ C×. Its structure is known for tirivial fiber groups A = 1 and A = C×. While it is a direct sum of simple biset functors in the case that A = 1, in the other case it is a simple C×-fibered biset functor. We noticed that as the fiber group grows, some of simple summands of 1-fibered biset functor CRC unite and form new fibered simple summands. In this thesis, we investigate the structure of the functor CRC for two intermediate fiber groups. The first one is a group containing all pn-th roots of unity for any n ∈ N and for any prime number p from a fixed set of primes π. The second one is the group of all pn-th roots of unity for a fixed n ∈ N. For both cases, we identify its new fibered simple summands by determining uniting summands via defining equivalence relations on them.Item Simple section biset functors(Thesis (Ph.D.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2022., 2022) Muslumov, Ruslan.; Coşkun, Olcay.Let G and H be finite groups and k be a commutative unitary ring. The Burnside group B(G,H) is the Grothendieck group of the category of finite (G,H)-bisets. The biset category kC of finite groups is the category defined over finite groups, whose morphism sets are given by the kB(G,H) groups. A biset functor defined on kC, with values in k-Mod is a k-linear functor from kC to the category of k-Mod. The remarkable results as the evaluation of the Dade group of endopermutation modules of a p- group and finding the unit group of the Burnside ring of a p- group are done using the theory of biset functors. Looking for ring objects in the category of biset functors one gets a more sophisticated structure, which is called a Green Biset Functor. Serge Bouc introduced the slice Burnside ring and the section Burnside ring for a finite group G. He also showed that these two rings have a natural structure of a Green Biset Functor. In our work we classify simple modules over the section Burnside ring of G using the approach of the paper Fibered Biset Functors by Robert Boltje and Olcay Coşkun.Item The fibered burnside ring of a fusion system(Thesis (Ph.D.) - Bogazici University. Institute for Graduate Studies in Science and Engineering, 2019., 2019.) Seviniş, Mert.; Coşkun, Olcay.J. S. Reeh, in his PhD thesis [1], described the Burnside ring and its free basis for the fusion-stable sets. In this thesis we will extend his results for fibered sets. Basically we will use the same technics J.S. Reeh developed in his thesis and we will show that these work with little modifications for the fibered case as well. In the end we hope to provide the reader with a similar description of the fıbered Burnside ring and also to write a free basis for fusion-stable fibered sets in general.