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Main menu for Browse IS/STAG
Course info
KMA / ALG
:
Course description
Department/Unit / Abbreviation
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KMA
/
ALG
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Academic Year
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2023/2024
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Academic Year
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2023/2024
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Title
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Algebraic Structures
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Form of course completion
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Exam
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Form of course completion
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Exam
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Accredited / Credits
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Yes,
5
Cred.
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Type of completion
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Combined
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Type of completion
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Combined
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Time requirements
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Lecture
4
[Hours/Week]
Seminar
1
[Hours/Week]
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Course credit prior to examination
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Yes
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Course credit prior to examination
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Yes
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Automatic acceptance of credit before examination
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No
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Included in study average
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YES
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Language of instruction
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Czech, English
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Occ/max
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|
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Automatic acceptance of credit before examination
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No
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Summer semester
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1 / -
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0 / -
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0 / -
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Included in study average
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YES
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Winter semester
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0 / -
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0 / -
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0 / -
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Repeated registration
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NO
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Repeated registration
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NO
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Timetable
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Yes
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Semester taught
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Summer semester
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Semester taught
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Summer semester
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Minimum (B + C) students
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1
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Optional course |
Yes
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Optional course
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Yes
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Language of instruction
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Czech, English
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Internship duration
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0
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No. of hours of on-premise lessons |
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Evaluation scale |
1|2|3|4 |
Periodicity |
každý rok
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Evaluation scale for credit before examination |
S|N |
Periodicita upřesnění |
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Fundamental theoretical course |
No
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Fundamental course |
No
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Fundamental theoretical course |
No
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Evaluation scale |
1|2|3|4 |
Evaluation scale for credit before examination |
S|N |
Substituted course
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None
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Preclusive courses
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N/A
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Prerequisite courses
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N/A
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Informally recommended courses
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N/A
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Courses depending on this Course
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KMA/AGM, KMA/ALS
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Histogram of students' grades over the years:
Graphic PNG
,
XLS
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Course objectives:
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The course aims at teaching the students to work with basic algebraic structures.
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Requirements on student
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Knowledge, understanding and aplications of algebraic structures.
Credit - individual assigment
Exam - orals: 3 topics (groups, rings and fields, reprezentations)
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Content
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Week 1: Groups, subgroups, Lagrange's Theorem.
Week 2: Normal subgroups, quotient groups.
Week 3: Homomorphisms of groups, theorems about isomorphism of groups.
Week 4: Cyclic groups and their structure, direct sum of groups.
Week 5: Chinese remainder theorem, groups of units modulo n
Week 6: Abelian Groups, direct decomposition of Abelian group and finite Abelian p-groups.
Week 7: Actions of groups, orbit counting theorem, groups of geometric transformations.
Week 8: Sylow p-subgroups and their properties.
Week 9: Chains of normal subgroups, Jordan-Holder theorem, solvable groups.
Week 10: Rings and fields, subrings, ideals, quotient rings, zero divisors, Euclidean rings.
Week 11: Rings of polynomials, splitting fields, solvability of polynomial equations in radicals, foundations of Galois theory.
Week12: Representations of finite groups. Regular representation, irreducible representation.
Week13: Character of representation.
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Activities
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Fields of study
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Guarantors and lecturers
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Literature
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-
Basic:
Abstraktní algebra
(Roman Nedela)
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Basic:
Joseph J. Rotman. An Introduction to the Theory of Groups. Springer Verlag, Berlin, 1995. ISBN 3-540-94285-8.
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Basic:
Joseph A. Gallian. Contemporary Abstract Algebra. Cengage, Boston, 2019. ISBN 978-1-305-65796-0.
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Basic:
Procházka, Ladislav. Úvod do studia reprezentací grup. 1. vyd. Praha : Karolinum, 1999. ISBN 80-246-0029-3.
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Extending:
Fuchs. Beskonečnyje Abelevy grupy. Mir, Moskva.
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Recommended:
Mac Lane, Saunders; Birkhoff, Garrett. Algebra. 2. vyd. Bratislava : Alfa, 1974.
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Recommended:
Algebra : Celost. vysokošk. učebnice pro stud. matematicko-fyzikálních a přírodovědeckých fakult, stud. oborů matematické vědy. 1. vyd. Praha : Academia, 1990. ISBN 80-200-0301-0.
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Recommended:
Lambek. Kolca i moduly. Mir Moskva.
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On-line library catalogues
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Time requirements
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All forms of study
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Activities
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Time requirements for activity [h]
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Contact hours
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65
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Graduate study programme term essay (40-50)
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45
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Preparation for an examination (30-60)
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50
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Total
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160
|
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Prerequisites
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Knowledge - students are expected to possess the following knowledge before the course commences to finish it successfully: |
ovládat základy lineární algebry v rozsahu předmětu KMA/LAA |
znát příklady konečných těles a jejich vlastnosti v rozsahu předmětu KMA/DMA |
pracovat s pojmy homomorfismus a isomorfismus v kontextu teorie grafů |
Basic knowledge in linear algebra and algebra is assumed. |
Skills - students are expected to possess the following skills before the course commences to finish it successfully: |
korektně formulovat matematická tvrzení |
používat základní důkazové techniky |
srozumitelně vysvětlit matematickou úvahu |
Competences - students are expected to possess the following competences before the course commences to finish it successfully: |
N/A |
N/A |
N/A |
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Learning outcomes
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Knowledge - knowledge resulting from the course: |
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umět aplikovat základní vlastnosti grup na konkrétní modely |
umět rozpoznat a analyzovat strukturu okruhu a tělesa |
umět rozpoznat počet ireducibilních reprezentací konečné grupy a navrhnout maticový tvar reprezentace grupy |
Skills - skills resulting from the course: |
nacházet souvislosti mezi teorií grup a dalšími matematickými teoriemi |
vysvětlit složitější argumenty v oblasti teorie algebraických struktur |
samostatně řešit problémy zaměřené na vlastnosti matematických struktur |
Competences - competences resulting from the course: |
N/A |
N/A |
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Assessment methods
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Knowledge - knowledge achieved by taking this course are verified by the following means: |
Combined exam |
Seminar work |
Individual presentation at a seminar |
Skills - skills achieved by taking this course are verified by the following means: |
Combined exam |
Seminar work |
Individual presentation at a seminar |
Competences - competence achieved by taking this course are verified by the following means: |
Combined exam |
Seminar work |
Individual presentation at a seminar |
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Teaching methods
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Knowledge - the following training methods are used to achieve the required knowledge: |
Interactive lecture |
Individual study |
Discussion |
Skills - the following training methods are used to achieve the required skills: |
Interactive lecture |
Individual study |
Discussion |
Competences - the following training methods are used to achieve the required competences: |
Interactive lecture |
Individual study |
Discussion |
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