Nielsen-Schreierov teorem

Semren, Matea (2014) Nielsen-Schreierov teorem. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

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.

Item Type: Thesis (Diploma thesis)
Supervisor: Perše, Ozren
Date: 2014
Number of Pages: 33
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 09 Jul 2015 12:38
Last Modified: 09 Jul 2015 12:38
URI: http://digre.pmf.unizg.hr/id/eprint/4106

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