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Lecturer(s)
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Kühr Jan, prof. RNDr. Ph.D.
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Chajda Ivan, prof. RNDr. DrSc.
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Kurač Zbyněk, Mgr.
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Course content
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Partitions and equivalence relations, quotient sets, mappings and equivalence relations. Groups (basic examples). Subgroups, subgroup generated by a subset, order of an element, order of a (sub)group. Left and right cosets, index of a subgroup, Lagrange?s theorem. Normal subgroups and quotient groups. Homomorphisms, normal subgroups and congruences, the homomorphism theorem. The center of a group, inner automorphisms. Symmetric groups, Cayley?s theorem. Cyclic groups, classification of cyclic groups. Direct products of groups, decomposition into direct product. Finite abelian groups. Rings, integral domains and division rings (basic examples). Subrings, ideals and quotient rings. Prime ideals and maximal ideals. Homomorphisms, ideals and congruences, the homomorphism theorem. The characteristic of a ring. Finite fields.
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Learning activities and teaching methods
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Lecture, Monologic Lecture(Interpretation, Training)
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Learning outcomes
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To understand the rudiments of the theory of groups and rings.
1. Knowledge Define basic notions, describe basic constructions and recall fundamental theorems of theory of groups and rings.
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Prerequisites
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unspecified
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Assessment methods and criteria
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Oral exam, Written exam
Credit: attendance at seminars, written test. Exam: oral exam, students have to demonstrate their knowledge and understanding of the subject matter.
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Recommended literature
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Grillet P. A. (2007). Abstract algebra. Springer New York.
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Milne J. S. Fields and Galois Theory.
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Milne J. S. Group Theory.
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Roman S. (2012). Fundamentals of Group Theory - An Advanced Approach. Birkhäuser.
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