Course information

Historical overview. Introduction to groups: Binary operation - equivalence relation. Groups, subgroups, homomorphisms-isomorphisms, symmetry groups, the n-th roots of unity, group structures with 2, 3, 4, 5 elements, quaternions. Cyclic groups and their classification. Permutation groups: Orbits, cycles, even and odd permutations, Cayley s Theorem. Homomorphisms and quotient groups: Cosets, Lagrange s Theorem, application to linear codes. Normal subgroup, quotient group, the Fundamental Theorem of homomorphisms. The commutator subgroup, abelianization. Free groups, group representation, topological applications. Free abelian groups, the classification of finitely generated abelian groups and their geometric interpretation. Group action on a set, Burnside s Theorem, applications to discrete mathematics problems. Introduction to rings, fields, integral domains and basic examples. Elements of number theory: Divisibility of integers, Euclidean algorithm, Bezout s Theorem. Congruences of integers, Fermat s and Euler s Theorems and applications, the Chinese remainder theorem, prime number theorems, unsolved problems and conjectures.