Algebra

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