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Course, academic year 2023/2024
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Introduction to Group Theory - NALG017
Title: Úvod do teorie grup
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018
Semester: winter
E-Credits: 6
Hours per week, examination: winter s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: cancelled
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. Mgr. Pavel Příhoda, Ph.D.
Classification: Mathematics > Algebra
Interchangeability : NMAG337
Is incompatible with: NMAG337
Is pre-requisite for: NALG010, NALG052, NALG033
Is interchangeable with: NMAG337
Annotation -
Free groups, defining relations, action of a group on a set, free, Cartesian and direct product, semidirect product, Abelian groups, finitely generated Abelian groups, Schreier's transversal and subgroups of free groups, Zassenhaus lemma, main and composition series, solvable groups and their characterization, Sylow theorems, nilpotent groups, characterization of nilpotent groups.
Last update: T_KA (20.05.2002)
Literature - Czech

1/ M.Aschbacher: Finite group theory, Cambridge University Press, 1986, 1988, 1993

2/ M.Hall: The theory of groups, Macmillan Company, New York, 1959 (též v ruském překladu)

3/ M.I.Kargapolov, Ju.I.Merzljakov: Osnovy teorii grup, Moskva, 1977

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

Last update: Zakouřil Pavel, RNDr., Ph.D. (05.08.2002)
Syllabus -

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.

Literature:

1/ M.Aschbacher: Finite group theory, Cambridge University Press, 1986, 1988, 1993

2/ M.Hall: The theory of groups, Macmillan Company, New York, 1959 (též v ruském překladu)

3/ M.I.Kargapolov, Ju.I.Merzljakov: Osnovy teorii grup, Moskva, 1977

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

Last update: T_KA (23.05.2002)
 
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