Jordan-Hölderov teorem za grupe

master's thesis

• Author: Mila Strpić
• Mentor: Ozren Perše (mentor)
• Committee member: Ozren Perše (predsjednik povjerenstva)
• Committee member: Marko Tadić (član povjerenstva)
• Committee member: Dražen Adamović (član povjerenstva)
• Committee member: Zrinka Franušić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-09-23, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj diplomski rad vezan je za grupe i neke teoreme vezane uz njih. Rad je podijeljen u dva poglavlja. Prvo poglavlje sastoji se od 4 potpoglavlja. U prvom potpoglavlju definiramo grupe i osnovna svojstva istih (potenciju elementa grupe i red grupe). U drugom potpoglavlju definiramo (prave) podgrupe, cikličke podgrupe, lijevu i desnu klasu te indeks podgrupe. Važan teorem u ovom potpoglavlju je Lagrangeov teorem. U trećem potpoglavlju definiramo homomorfizam, normalne podgrupe i centar grupe. U četvrtom potpoglavlju proučavamo normalne podgrupe i kvocijentne grupe. Također iskazujemo i dokazujemo tri teorema o izomorfizmu i definiramo proste grupe. Drugo poglavlje nosi naziv kao i sam rad. Na početku poglavlja iskazujemo i dokazujemo Zassenhausovu ili leptir-lemu, definiramo normalni i kompozicijski niz, definiramo ekvivalentnost dva normalna niza grupe, profinjenje normalnog niza. Iskazujemo i dokazujemo Schreierov teorem o profinjenju te kao njegovu posljedicu dokazujemo Jordan-Hölderov teorem, koji je jedan od centralnih rezultata rada. Također definiramo komutatorske podgrupe i središnji padajući niz grupe.

Abstract (english)

This diploma thesis is devoted to the groups and some theorems from group theory. The thesis is divided into two chapters. The first chapter consists of four sections. In the first section we define the group and some basic properties of groups (powers of group elements and group order). In the second section we define (proper) subgroups, cyclic subgroups, left and right cosets and index of a subgroup. An important theorem in this subsection is Lagrange’s theorem. In the third section we define a homomorphism, normal subgroups and the center of a group. In the fourth section we study normal subgroups and quotient groups. We state and prove three theorems on isomorphisms and define simple groups. The second chapter is titled as the work itself. At the beginning of this chapter we state and prove Zassenhaus or butterfly-lemma, we define normal and composition series, we define the equivalence of two normal series of the group and refinement of the normal sequence. We state and prove Schreier’s refinement theorem and as its consequence, we obtain the Jordan-Hölder theorem, which is one of the main results of thif thesis. We also define a comutator subgroups and central descending series of a group.

• Keywords: Jordan-Hölderov teorem; grupe; podgrupe; Lagrangeov teorem; homomorfizam; izomorfizam; Zassenhausovu ili leptir-lema; Schreierov teorem o profinjenju
• Keywords (english): Jordan-Hölderov theorem; groups; subgroups; Lagrange’s theorem; homomorphism; isomorphisms; Zassenhaus or butterfly-lemma; Schreier’s refinement theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:906961
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Access restricted to students and staff of home institution
• Created on: 2019-02-08 10:43:36