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Lecturer(s)
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Plešinger Martin, doc. Ing. Ph.D.
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Course content
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1st part - Basics of group theory Prerequisites: Sets, properties of set operations. Cartesian product, binary relations, their properties. Partition and equivalence on a set. Binary (n-ary) operations and their properties. C-tables. Algebraic structures. Functions, properties of function composition. Inverse function. {in; sur; bi}jection. 1. Cyclic groups, generator. Cycle graph of a group. Additive and multiplicative groups of modular arithmetic. 2. Finite groups. Lagrange theorem. Group presentation. 3. Small groups and their structure. Subgroups. Inclusive graph of (normal) subgroups. 4. Direct product of groups. Groups and symmetry. Dihedral groups. 5. Permutation groups. Symmetric a alternating groups (esp. permutations on 3, or 4 entries). 6. Cosets, kernel and image. Normal subgroups. Factor groups. 7. Morphism, special cases. {Iso; homo; auto}morphism of groups. Homomorphism, its kernel, and normal subgroup. 8. Rings. Ideal {one; two}-sided. Maximal and prime ideal. Factor rings. 2nd part - Basics of lattice theory and of Boolean algebra 9. Partially ordered sets (= posets), join and meet. Principle of duality. Inclusion diagram. 10. Lattice identities. Sublattices, product of lattices. 11. Modular lattices. Distributive lattices, set rings. 12. Boolean lattices, Boolean algebra. 13. Application of Boolean algebra. Minimization of B-functions. 14. Time reserve. Continuously - applications of Maple environment in group theory.
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Learning activities and teaching methods
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Monological explanation (lecture, presentation,briefing)
- Class attendance
- 42 hours per semester
- Preparation for credit
- 14 hours per semester
- Preparation for exam
- 28 hours per semester
- Home preparation for classes
- 36 hours per semester
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Learning outcomes
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Prohloubení základních znalostí obecné algebry a matematických struktur. Struktury s jednou a dvěma operacemi, uspořádané struktury, obecné algebry. Úvod do pokročilejších patrií: Liovy grupy, kvaterniony a oktorniony, boolovské algebry.
Theory, algorithms and applications of theory of groups, lattices, and Boolean algebra. Applications in geometry, symmetry, and synthesis of logical schemas.
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Prerequisites
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Algebra and geometry 1, 2.
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Assessment methods and criteria
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Combined examination, Oral exam, Written exam
Oral presentation of semester work in seminar (small groups of order at most 15, isomorphy of concrete models, applications).
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Recommended literature
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Beachy, J.A. - Blair, W.D. Abstract algebra, 3rd Edit.. Waveland Press, Inc., 2005.
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Bican, L. Algebra (pro učitelské studium). Praha, Academia, 2001. ISBN 80-200-0860-8.
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Birkhoff, G. - Bartee, T.C. Aplikovaná algebra. Bratislava, Alfa, 1981.
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Gallian, J. Contemporary Abstract Algebra. Boston, N.Y., Houlinghton Mifflin Comp., 1998. ISBN 0-618-51471-6.
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Gilbert J.:. Elements of modern algebra. PWS-KENT Publishing Company, Boston, 1988.
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Katriňák, T. aj. Algebra a teoretická aritmetika (1). Blava/Praha, Alfa/SNTL, 1985.
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Kopka J.:. Svazy a Booleovy algebry. Ústí nad Labem, UJEP, 1991. ISBN 80-7044-025-2.
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Mac Lane, S. - Birkhoff, G. Algebra. Bratislava, Alfa, 1973.
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Vild, J. - Šedý, J. Matematika II [Algoritmy a logika]. Liberec, VŠST, 1978.
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Weil, J. aj. Rozpracovaná řešení úloh z vyšší algebry. Praha, 1987.
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