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Algebra

* Exchange students do not have to consider this information when selecting suitable courses for an exchange stay.

Course Unit Code470-2204/03
Number of ECTS Credits Allocated6 ECTS credits
Type of Course Unit *Compulsory
Level of Course Unit *First Cycle
Year of Study *Third Year
Semester when the Course Unit is deliveredWinter Semester
Mode of DeliveryFace-to-face
Language of InstructionCzech
Prerequisites and Co-Requisites Course succeeds to compulsory courses of previous semester
Name of Lecturer(s)Personal IDName
KOV16doc. Mgr. Petr Kovář, Ph.D.
JAH02RNDr. Pavel Jahoda, Ph.D.
Summary
Selected topics of general algebra constitute content of course Algebra. Possibilities of using this knowledges to solve some practical problems are demonstrated here. Students have the opportunity to obtain basic familiarity with mathematical apparatus, which stands behind the above mentioned applications. So they can understand how these applications work in practice.
Learning Outcomes of the Course Unit
After passing the course the student will be familiar with the definitions of basic concepts selected theory of algebraic structures and relationships between them. He will understand their significance and will be able to take advantage of their knowledge to solve simple algebraic structures theory tasks. He will also understand the importance of these concepts for the solution of the selected application roles, so that he could formulate a practical role in the language of group theory, solve the problem using theory and tools to interpret the outcome in the context of the original task.
Course Contents
Lectures
1) introduction to the group theory: symmetry and dihedral groups
2) group: definition, basic properties
3) finite groups and subgroups, examples
4) cyclic groups, classification
5) group of permutations, definitions, cycles, properties and use
6) normal subgroups and Lagrange's theorem
7) factor groups
8) homomorphisms of groups, definitions, examples
9) isomorfisms: motivation, properties, Cayley's theorem
10) direct product of groups, definitions, examples, applications
11) rings and fields: definitions, finite and infinite examples, applications
12) fields, algebraic extensions, examples, applications
13) vector spaces: definition and examples, subspaces, linear independence

Cvičení:
1) examples of dihedral groups, geometric meaning, examples
2) examples of groups, verification of the axioms of groups
3) subgroups, examples, design and verification
4) cyclic groups, examples, properties, verification
5) group of permutations, cycles, solving the practical examples
6) factorisation the group by its subgroup
7) examples of factor groups, construction and verification
8) homomorfisms of groups, definitions, examples
9) isomorfisms, examples and counterexamples, verification of axioms
10) direct product of groups, examples
11) homomorfisms og groups
12) rings and fields: examples, verification
13) vector spaces: finite and infinite examples, verification of linear independence
Recommended or Required Reading
Required Reading:
J. GALLIAN: Contemporary Abstract Algebra, Cengage Learning; 8 edition (2012), ISBN13 978-1133599708.
G. BIRKHOFF, S. MAC LANE: Algebra, Alfa, Bratislava 1974.

J. GALLIAN: Contemporary Abstract Algebra, Cengage Learning; 8 edition (2012), ISBN13 978-1133599708.
Recommended Reading:
J. Gallian, Contemporary Abstract Algebra, Cengage Learning; 8 edition (2012), ISBN13 978-1133599708.

LANG, S.: Undergraduate Algebra, Springer, 1990, ISBN 0-387-97279-X.
GALLIAN, J.: Contemporary Abstract Algebra, Houghton Mifflin, Boston 2002, ISBN 0-618-122141
J. GALLIAN: Contemporary Abstract Algebra, Cengage Learning; 8 edition (2012), ISBN13 978-1133599708.

BIRKHOFF, G., S.MAC LANE: Algebra, Alfa, Bratislava 1974.
Planned learning activities and teaching methods
Lectures, Tutorials
Assesment methods and criteria
Task TitleTask TypeMaximum Number of Points
(Act. for Subtasks)
Minimum Number of Points for Task Passing
Credit and ExaminationCredit and Examination100 (100)51
        CreditCredit30 15
        ExaminationExamination70 35