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Course, academic year 2019/2020
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Introduction to Group Theory - NMAG337
Title in English: Úvod do teorie grup
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2019
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2 C+Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: Czech
Teaching methods: full-time
Guarantor: doc. Mgr. et Mgr. Jan Žemlička, Ph.D.
Class: M Bc. MMIT
M Bc. MMIT > Doporučené volitelné
M Bc. OM
M Bc. OM > Zaměření MSTR
M Bc. OM > Povinně volitelné
M Mgr. MMIB > Volitelné
Classification: Mathematics > Algebra
Incompatibility : NALG017
Interchangeability : NALG017
In complex pre-requisite: NMAG349, NMAG351
Annotation -
Last update: G_M (15.05.2012)
A recommended course on group theory for specialization Mathematical Structures within General Mathematics.
Course completion requirements - Czech
Last update: Mgr. Jan Šaroch, Ph.D. (07.10.2018)

Zápočet se uděluje za aktivní účast na cvičeních. Zápočet není nutnou podmínkou účasti u zkoušky.

Literature -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (02.10.2012)

Aleš Drápal: Teorie grup : základní aspekty, Karolinum, Praha, 2000.

Derek J.S. Robinson: A Course in the Theory of Groups, Springer, New York, 1982.

Joseph J. Rotman: An Introduction to the Theory of Groups, Springer, New York, 1995.

M. Hall: The Theory of Groups, Macmillan Company, New York, 1959.

I.Martin: Isaacs, Finite group theory, American Mathematical Society, Providence, 2008.

L. Procházka, L. Bican, T. Kepka, P. Němec: Algebra, Academia, Praha, 1990.

Requirements to the exam - Czech
Last update: Mgr. Jan Šaroch, Ph.D. (07.10.2018)

Zkoušená témata vycházejí z látky probrané na přednášce a cvičeních; důraz bude kladen především na zvládnutí teorie. Zkouška se koná ústní formou.

Syllabus -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (02.10.2012)

1. Free base, free groups, reduced words.

2. Defining relations. Examples.

3. Group actions on a set. Actions by translations and conjugations. The kernel of an action.

4. Free product and its reduced words.

5. Cartesian and direct products. Characterization by normal subgroups.

6. Semidirect product and its structural meaning. Examples.

7. Abelian groups - product and coproduct. Finitely generated abelian groups. Cardinality of the basis of a free group.

8. Schreier's transversal and subgroups of a free group.

9. Zassenhaus lemma. Main and composition series.

10. Solvable groups, closeness for factors etc. Description by normal aand subnormal series.

11. Sylow theorems.

12. Upper and lewer central series. Nilpotent groups. Description of finite nilpotent groups.

The simplicity of the alternating groups will be proved in the exercise classes. Characterization of divisible groups is proved when it is not included in the concurrent lecture on module theory.

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