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Course, academic year 2023/2024
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Combinatorial Group Theory - NMAG431
Title: Kombinatorická teorie grup
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: summer
E-Credits: 6
Hours per week, examination: summer s.:3/1, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. Mgr. Pavel Růžička, Ph.D.
Class: M Mgr. MSTR
M Mgr. MSTR > Povinně volitelné
Classification: Mathematics > Algebra
Incompatibility : NALG033
Interchangeability : NALG033
Is co-requisite for: NMAG432
Annotation -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (14.05.2020)
Subgroups of free groups (Nielsen's and Reidemaister's method), Tietze transformations, HNN extensions, free products with an amalgamated subgroup, geometrical methods, Cayley complexes. Other selected topics in elementary combinatorical group theory.
Course completion requirements -
Last update: doc. Mgr. Pavel Růžička, Ph.D. (29.10.2019)

Students are assumed to attend both the winter and the summer semestr. The exam will be in summer.

Literature -
Last update: doc. Mgr. Pavel Růžička, Ph.D. (10.10.2017)

1. Rotman, J. J., An Introduction to The Theory of Groups (2nd ed.), Springer-Verlag, 1999.

2. Lyndon, R. C. and Schupp, P. E., Combinatorial Group Theory (Reprint of the 1977 ed.), Springer-Verlag, Berlin Heilderberg NY, 2001.

3. Magnus, W., Karrass, A., Solitar, D., Combinatorial Group Theory (Representation of Groups in Generators and Relations), Dower Publ. INC, Mineola NY, 2004.

4. Bogopolski, O., Introduction to Group Theory (EMS Textbooks in Mathematics, EMS Publ. House, Zurich, Switzerland, 2008.

Syllabus -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (14.05.2020)

Basics of combinatorial group theory:

1. Free group, subgroups of a free group (the method of Nielsen and Reidemeister), the relationship between the index and the rank of subgroup of a group of a finite index, subgroups of finite rank as free factors in a subgroup of a finite index. Conjugation and cyclically reduced words.

2. Tietze transformations.

3. HNN extensions, defining relations, Britton's lemma and the normal form theorem, applications of HNN extensions.

4. Free products with an amalgamated subgroup, defining relations, the normal form theorem.

5. Geometrical methods, the fundamental group of a two-dimensional complex, application for a geometrical proof that a subgroup of a free group is free, Kurosh's theorem, Grushko -- von Neumann's theorem.

6. Cayley complexes

According to an interest, some of the following topics will be tought.

1. Higman's embedding theorem.

2. Small cancellation theory.

3. Braid group, the word problem, factors, connections to authomorphisms of free groups.

4. Groups acting on trees.

5. Hyperbolic groups.

6. Tessellations and Fuchsian complexes.

7. Solvability of the word problem for groups with one defining relation.

8. Bipolar structures.

Entry requirements -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (17.05.2019)

Basics of group theory.

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