| Course title | Algebra Seminar |
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| Course code | KMT/SAL@ |
| Organizational form of instruction | Seminar |
| Level of course | Bachelor |
| Year of study | not specified |
| Semester | Summer |
| Number of ECTS credits | 3 |
| Language of instruction | Czech |
| Status of course | Compulsory-optional |
| Form of instruction | Face-to-face |
| Work placements | This is not an internship |
| Recommended optional programme components | None |
| Lecturer(s) |
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| Course content |
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Course Content - From Definition to Image: Introduction to Group Explorer. Cayley diagrams as representations of group structure. Selecting generators and their impact on the resulting graph. - Symmetry in Space: Visualizing dihedral groups ($D_n$) and symmetric groups ($S_n$). Finding internal connections between permutations and geometric intuition. - Subgroups and their Hierarchy: Visual identification of subgroups in Cayley diagrams and lattice diagrams. - Cosets and Lagrange's Theorem: How to "see" cosets. Visual proof of Lagrange's theorem by partitioning the diagram into disjoint regions. - Normal Subgroups and Quotient Groups: Visualizing the most difficult concepts. Organizing diagrams by cosets and recognizing the "regularity" that defines normality. - Homomorphisms: Mapping between groups. Visualizing the kernel and the image using color coding in Group Explorer. - Advanced Structures: Direct and semidirect products - how complex shapes are formed by combining simpler ones.
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| Learning activities and teaching methods |
Monologic Lecture(Interpretation, Training)
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| Learning outcomes |
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The aim of the course is to transform abstract knowledge of algebraic structures into illustrative visual models. Students will learn to use the Group Explorer software to demonstrate group properties, enabling them to better explain symmetries and abstract relationships in their future teaching practice.
Upon successful completion of the course, the student will have gained deep insight into the structure of abstract groups through visual intuition and modern software tools. The graduate will be able to: Visual Analysis: Interpret and independently create complex visual models of groups (Cayley diagrams, multiplication tables, subgroup lattices) within the Group Explorer environment. Didactic Transformation: Transform abstract algebraic theorems (e.g., Lagrange's theorem or Isomorphism theorems) into clear visual representations suitable for teaching at various educational levels. Analytical Skills: Identify key group properties (commutativity, order of elements, normality of subgroups) directly from their geometric representation. Academic Literacy: Effectively utilize professional English literature (N. Carter: Visual Group Theory) to design innovative teaching materials. Digital Competence: Master specialized mathematical software and integrate it into the educational process as a tool for discovery and experimentation. |
| Prerequisites |
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Successfully completed the course Algebra 1 (Group Theory).
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| Assessment methods and criteria |
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Student performance
To successfully complete the course, students must fulfill the following requirements: A. Software Project (Term Project)The student will submit an interactive portfolio created in Group Explorer (or a set of exported diagrams with commentary), which must include: - A custom visualization of a selected group (e.g., the quaternion group Q_8 or the alternating group A_4). - A graphical representation of all its proper subgroups. - A visualization of at least one non-trivial homomorphism (e.g., from S_3 to C_2). B. Literature Study The student will select one chapter from the book Visual Group Theory (N. Carter) and prepare a 15-minute pedagogical presentation for their peers. In this presentation, they will explain a specific abstract concept (e.g., "Normality") using purely visual arguments from the book and software, avoiding the classical "formal/rigorous" style of proof notation. C. Final Test (Interpretation)A short assessment where the student's task is not to perform calculations, but to interpret an unknown diagram. (Example: "Based on this Cayley diagram, determine whether the group is Abelian and identify its center.") |
| Recommended literature |
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| Study plans that include the course |
| Faculty | Study plan (Version) | Category of Branch/Specialization | Recommended semester | |
|---|---|---|---|---|
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB19) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB25) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB24) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB22) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB22) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB25) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB24) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB23) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB20) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB19) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB21) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB20) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics focused on education (BB21) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |
| Faculty: Faculty of Education | Study plan (Version): Mathematics: teaching focus (BB23) | Category: Pedagogy, teacher training and social care | 3 | Recommended year of study:3, Recommended semester: Summer |