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Contributions to projective group theory
Thesis (MSc)--Stellenbosch University, 2017
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2017
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| _version_ | 1867614096277372928 |
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| access_status_str | Open Access |
| author | Van Niekerk, Francois Koch |
| author2 | Janelidze, Zurab |
| author_browse | Janelidze, Zurab Van Niekerk, Francois Koch |
| author_facet | Janelidze, Zurab Van Niekerk, Francois Koch |
| author_sort | Van Niekerk, Francois Koch |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc)--Stellenbosch University, 2017 |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/102912 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:46:36.532Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2017 |
| publishDateRange | 2017 |
| publishDateSort | 2017 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/102912 Contributions to projective group theory Van Niekerk, Francois Koch Janelidze, Zurab Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Projective linear groups Group theory Linear algebraic groups Commutators (Mathematics) UCTD Thesis (MSc)--Stellenbosch University, 2017 ENGLISH ABSTRACT : Projective Group Theory (PGT for short) provides a self-dual axiomatic context that allows one to establish homomorphism theorems for (non-abelian) group-like structures. The present thesis has two broad aims. The rst is to introduce a "norm function" in PGT as a way to capture the notion of an order of a (finite) group in PGT, extend some elementary results on nite groups to PGT, propose a de nition of a ( nite) cyclic group in PGT, and make an attempt to recapture the Second Sylow Theorem. We also describe a process of building a model for normed PGT (i.e. PGT with a norm function), from a monoid equipped with a family of congruences, subject to suitable axioms. In the case of the multiplicative monoid of natural numbers equipped with the family of modular congruences, we recover the model of normed PGT formed by nite cyclic groups. The second aim of the thesis is to introduce and study biproducts and commutators in PGT, which generalize usual products and commutators for group-like structures. Our biproducts are not categorical products, although, as we show, they form a monoidal structure. However, our notion of a biproduct is self-dual, just like (and is in fact very similar to) the one in the context of an abelian category. AFRIKAANSE OPSOMMING : Projektiewe groepteorie (PGT vir kort) bied 'n selfduale konteks wat mens toelaat om homomor sme stellings vir (nie-abelse) groepagtige strukture vas te stel. Die huidige tesis het twee bre e doelwitte. Die eerste is om 'n "norm funksie" in PGT voor te stel as 'n manier om die konsep van 'n orde van 'n (eindige) groep in PGT vas te vang, sommige element^ere resultate van eindige groepe na PGT te verleng, 'n de nisie van (eindige) sikliese groepe in PGT voor te stel, en om 'n poging aan te wend om die Tweede Sylow Stelling vas te vang. Ons beskryf ook 'n proses om modelle vir genormeerde PGT (d.w.s. PGT met 'n norm funksie) te bou van mono ede toegerus met 'n familie van kongruensies, onderhewig aan geskikte aksiomas. In die geval van die multiplikatiewe mono ed van natuurlike getalle toegerus met die familie van modul^ere kongruensies, kry ons die model van genormeerde PGT gevorm deur eindige sikliese groepe terug. Die tweede doelwit van hierdie tesis is om biprodukte en kommutators in PGT voor te stel en te studeer, wat die gewone produkte en kommutators vir groepagtige strukture veralgemeen. Ons biprodukte is nie kategoriese produkte nie, alhoewel, soos ons wys, vorm hulle 'n monoïedale struktuur. Egter, ons konsep van biproduk is selfduaal, net soos (en is in werklikheid baie soortgelyk tot) die een in die konteks van 'n abelse kategorie. 2017-11-27T13:58:26Z 2017-12-11T11:11:23Z 2017-11-27T13:58:26Z 2017-12-11T11:11:23Z 2017-12 Thesis http://hdl.handle.net/10019.1/102912 en_ZA Stellenbosch University v, 67 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Projective linear groups Group theory Linear algebraic groups Commutators (Mathematics) UCTD Van Niekerk, Francois Koch Contributions to projective group theory |
| title | Contributions to projective group theory |
| title_full | Contributions to projective group theory |
| title_fullStr | Contributions to projective group theory |
| title_full_unstemmed | Contributions to projective group theory |
| title_short | Contributions to projective group theory |
| title_sort | contributions to projective group theory |
| topic | Projective linear groups Group theory Linear algebraic groups Commutators (Mathematics) UCTD |
| url | http://hdl.handle.net/10019.1/102912 |
| work_keys_str_mv | AT vanniekerkfrancoiskoch contributionstoprojectivegrouptheory |