Teorija konačnih grupa

master's thesis

• Title (english): Finite group theory
• Author: Martina Bilić
• Mentor: Dražen Adamović (mentor)
• Committee member: Dražen Adamović (predsjednik povjerenstva)
• Committee member: Hrvoje Planinić (član povjerenstva)
• Committee member: Franka Miriam Brückler (član povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2022-06-01, 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: monoidi; Schreierov teorem; Jordan Hoelderov teorem; jednadžba klasa; Sylowljeve grupe; Abelove grupe
• Keywords (english): monoids; Schreier theorem; Jordan Hoelder theorem; the class formula; Sylow subgroups; Abelian groups
• Language: croatian
• URN:NBN: urn:nbn:hr:217:387773
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2022-06-07 08:31:15