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master's thesis
Nielsen-Schreierov teorem
Zagreb: University of Zagreb, Faculty of Science, 2014. urn:nbn:hr:217:954634
Semren, Matea
University of Zagreb
Faculty of Science
Department of Mathematics

Institutional repository: Repository of the Faculty of Science

Cite this document

Semren, M. (2014). Nielsen-Schreierov teorem (Master's thesis). Zagreb: University of Zagreb, Faculty of Science. Retrieved from https://urn.nsk.hr/urn:nbn:hr:217:954634

Semren, Matea. "Nielsen-Schreierov teorem." Master's thesis, University of Zagreb, Faculty of Science, 2014. https://urn.nsk.hr/urn:nbn:hr:217:954634

Semren, Matea. "Nielsen-Schreierov teorem." Master's thesis, University of Zagreb, Faculty of Science, 2014. https://urn.nsk.hr/urn:nbn:hr:217:954634

Semren, M. (2014). 'Nielsen-Schreierov teorem', Master's thesis, University of Zagreb, Faculty of Science, accessed 26 December 2024, https://urn.nsk.hr/urn:nbn:hr:217:954634

Semren M. Nielsen-Schreierov teorem [Master's thesis]. Zagreb: University of Zagreb, Faculty of Science; 2014 [cited 2024 December 26] Available at: https://urn.nsk.hr/urn:nbn:hr:217:954634

M. Semren, "Nielsen-Schreierov teorem", Master's thesis, University of Zagreb, Faculty of Science, Zagreb, 2014. Available at: https://urn.nsk.hr/urn:nbn:hr:217:954634

Metadata
TitleNielsen-Schreierov teorem
AuthorMatea Semren
MentorOzren Perše (mentor)
Committee memberOzren Perše (predsjednik povjerenstva)
Committee memberMirko Primc (član povjerenstva)
Committee memberZvonimir Bujanović (član povjerenstva)
Committee memberLjiljana Arambašić (član povjerenstva)
GranterUniversity of Zagreb
Faculty of Science
(Department of Mathematics)
Zagreb
Defense date and country2014-09-24, 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
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:217:954634
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 on2019-02-06 14:16:10