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master's thesis
Teorija konačnih grupa
Zagreb: University of Zagreb, Faculty of Science, 2022. urn:nbn:hr:217:387773
Bilić, Martina
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Bilić, M. (2022). Teorija konačnih grupa (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:387773

Bilić, Martina. "Teorija konačnih grupa." Master's thesis, University of Zagreb, Faculty of Science, 2022. https://urn.nsk.hr/urn:nbn:hr:217:387773

Bilić, Martina. "Teorija konačnih grupa." Master's thesis, University of Zagreb, Faculty of Science, 2022. https://urn.nsk.hr/urn:nbn:hr:217:387773

Bilić, M. (2022). 'Teorija konačnih grupa', Master's thesis, University of Zagreb, Faculty of Science, accessed 03 July 2024, https://urn.nsk.hr/urn:nbn:hr:217:387773

Bilić M. Teorija konačnih grupa [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2022 [cited 2024 July 03] Available at: https://urn.nsk.hr/urn:nbn:hr:217:387773

M. Bilić, "Teorija konačnih grupa", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2022. Available at: https://urn.nsk.hr/urn:nbn:hr:217:387773

Metadata
TitleTeorija konačnih grupa
Title (english)Finite group theory
AuthorMartina Bilić
MentorDražen Adamović (mentor)
Committee memberDražen Adamović (predsjednik povjerenstva)
Committee memberHrvoje Planinić (član povjerenstva)
Committee memberFranka Miriam Brückler (član povjerenstva)
Committee memberMarko Vrdoljak (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2022-06-01, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Abstract
Glavni cilj ovog rada bio je dati sažeti prikaz teorije grupa. U prvom poglavlju smo iznijeli osnovne pojmove i rezultate vezane za monoide, da bismo pomoću njih u drugom poglavlju definirali grupe. Kroz poglavlje o grupama, osim proučavanja njihovih osnovnih svojstava, definirali smo i normalne podgrupe te nizove podgrupa i rješive grupe. Iskazali smo Schreierov te Jordan Hoelderov teorem. Nadalje, definirali smo cikličke grupe te kroz iskaze propozicija proučili njihova svojstva. Definirali smo i djelovanje grupe na skupu gdje smo saznali više o algebarskoj strukturi grupe. Taj dio smo zaključili teoremom o jednadžbi klasa. Promotrili smo Sylowljeve grupe te smo nakon definiranja osnovnih pojmova vezanih uz njih, iskazali i Sylowljeve teoreme. U trećem poglavlju prezentirali smo Abelove grupe. Definirali smo pojam direktne sume, baze te slobodne Abelove grupe. Kod proučavanja konačno generiranih Abelovih grupa, posebnu pažnju posvetili smo torzionoj podgrupi koju smo definirali kao skup svih elemenata grupe konačnog reda. U četvrtom poglavlju naveli smo neke primjere konačnih grupa i neke strukturne teoreme. Definirali smo diedralnu grupu i poludirektan produkt, a kroz propozicije smo dobili uvid u osnovna svojstva grupa reda \(p^2 \), odnosno reda \(pq \). Za kraj smo klasificirali grupe reda 8 i 12.
Abstract (english)
The aim of this master’s thesis was to provide a systematic review of theory of finite groups. The first chapter begined with basic notions and results of monoids, which introduced the term of groups in the second chapter. Through the chapter about groups, besides study of their basic properties, we defined normal subgroups, tower of subgroups and solvable groups. We presented Schreier and Jordan Hoelder theorem. Furthermore, we presented operation of a group on a set where we found out more about algebraic structure of group. We concluded that part with the class formula. We considered Sylow subgroups and after defining basic notions regarding them, we presented Sylow theorems. In the third chapter, we analized Abelian groups. We defined direct sum, basis and free Abelian group. When studying finitely generated Abelian groups, we paid special attention to torsion subgroup which we defined as set of all elements of group with finite period. In the fourth chapter, we stated some examples of finite groups and some structural theorems. We defined dihedral group and semidirect product and through propositions we gave an insight in basic properties of groups of order \(p^2 \) and groups od oreder \(pq \). In the end we classified groups of order 8 and 12.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:387773
Study programmeTitle: Finance and Business Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2022-06-07 08:31:15