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Course, academic year 2022/2023
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Introduction to Group Theory - NMAG337
Title: Úvod do teorie grup
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2022
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Additional information: http://www1.karlin.mff.cuni.cz/~kala/web/teaching/grupy21
Guarantor: doc. Mgr. Pavel Růžička, 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
M Mgr. MMIB > Volitelné
Classification: Mathematics > Algebra
Incompatibility : NALG017
Interchangeability : NALG017
Is interchangeable with: 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 -
Last update: doc. Mgr. Vítězslav Kala, Ph.D. (15.09.2021)

Credit will be awarded for succesfully solving several homework sets (see web for details). The nature of the assessment of study allows for repeated attempts at earning credit.

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.

Teaching methods - Czech
Last update: doc. Mgr. Vítězslav Kala, Ph.D. (15.09.2021)

Prozatím plánujeme prezenční výuku (se striktním dodržováním platných koronavirových nařízení).

Requirements to the exam -
Last update: doc. Mgr. Vítězslav Kala, Ph.D. (15.09.2021)

Students have to pass final oral exam. The requirements for the exam correspond to what has been done during lectures and practicals.

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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