Nielsen-Schreierov teorem

master's thesis

• Author: Matea Semren
• Mentor: Ozren Perše (mentor)
• Committee member: Ozren Perše (predsjednik povjerenstva)
• Committee member: Mirko Primc (član povjerenstva)
• Committee member: Zvonimir Bujanović (član povjerenstva)
• Committee member: Ljiljana Arambašić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2014-09-24, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu dokazujemo jedan od fundamentalnih rezultata o slobodnim grupama. To je Nielsen-Schreierov teorem, koji kaže da je svaka podgrupa slobodne grupe također slobodna grupa. Najprije ćemo definirati grupu, podgrupu, generiranu podgrupu, normalnu grupu, te kvocijentne grupe koje nam govore kako iz već postojeće grupe konstruirati novu grupu. Također ćemo definirati homomorfizam između grupa, i posebno izomorfizam, koji nam omogućuju da usporedimo grupe, te dokazati tri teorema o izomorfizmima. U drugom ćemo dijelu najprije definirati direktnu sumu Abelovih grupa preko koje ćemo definirati slobodnu Abelovu grupu i njenu bazu. Karakterizacija slobodnih Abelovih grupa nam je motivacija za općenitu definiciju slobodnih grupa. Postojanje ne-abelove slobodne grupe dokazat ćemo tako što ćemo je konstruirati iz proizvoljnog nepraznog skupa, koji će biti baza za tu slobodnu grupu. Zatim ćemo definirati transverzalu i posebno Schreierovu transverzalu preko koje definiramo bazu za podgrupu slobodne grupe. Na kraju ćemo navesti nekoliko posljedica Nielsen-Schreierog teorema.

Abstract (english)

In this thesis we prove one of the fundamental results about free groups. That is the Nielsen-Schreier theorem, which says that every subgroup of free group is also free group. First, we define a group, a subgroup, a generated subgroup, a normal subgroup and quotient groups which allows us to construct a new group from given group. We will also define a homomorphism between groups, and specially an isomorphism, and prove the three isomorphism theorems. In second part, first, we define a direct sum, which allows us to define a free abelian group and its basis. The characterisation of free abelian groups is our motivation for more general definition of free groups. We prove existence of non-abelian free group by constructing it from some nonempty set, which is a basis for that free group. Then, we define a transversal, and specially Schreier transversal, which allows us to define a basis for a subgroup of free group. In the end, we give some applications of the Nielsen-Schreier theorem.

• Keywords: Nielsen-Schreierov teorem; slobodne grupe; homomorfizam; izomorfizam; Abelove grupe; Schreierova transverzala
• Keywords (english): Nielsen-Schreier theorem; free group; homomorphism; isomorphism; abelian groups; Schreier transversal
• Language: croatian
• URN:NBN: urn:nbn:hr:217:954634
• 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: 2019-02-06 14:16:10