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Lecturer(s)
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Dofková Radka, doc. PhDr. Ph.D.
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Zdráhal Tomáš, doc. RNDr. CSc.
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Course content
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Contents Introduction to Group Explorer software and Cayley diagrams as representations of group structure. Symmetry in space, visualization of dihedral and symmetric groups, relationship between permutations and geometry. Subgroups and their hierarchy, identification in lattice diagrams. Cosets and Lagrange's theorem, visual proofs using disjoint regions. Normal subgroups and factor groups, organization of diagrams according to cosets and recognition of structural regularity. Homomorphisms, visualization of kernels and images using color coding. Advanced structures, direct and semidirect products as combinations of simpler forms.
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Learning activities and teaching methods
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unspecified
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Learning outcomes
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Course objectives The aim of the course is to provide students with an intuitive and visual insight into group theory. Instead of a purely axiomatic approach, emphasis is placed on geometric imagination and visualization of abstract structures. Students will learn to understand groups as actions on objects and to represent them using Cayley diagrams, operation tables, and lattice diagrams. Another important objective is to learn how to use the specialized software Group Explorer for experimental exploration of algebraic properties
Competencies (Learning outcomes) After completing the course, students will: Create and interpret Cayley diagrams for different types of groups and understand the influence of the choice of generators on their form. Visually identify subgroups, cosets, and normal subgroups in diagrams. Geometrically interpret dihedral and symmetric groups. Demonstrate the validity of Lagrange's theorem and the structure of factor groups using visual decompositions. Effectively uses Group Explorer software to analyze group properties, find cores, and find images of homomorphisms.
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Prerequisites
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unspecified
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Assessment methods and criteria
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unspecified
50% attendance tutorial work The course will take place according to the schedule that the student individually agrees with the teacher.
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Recommended literature
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Haviar, M, Klenovčan, P. Basic algebra for future teachers. .
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