Course program
Sets
Correspondences. Mappings. Composition of mappings. Associativity of composition. Injective, surjective e bijective mappings. Inverse mapping of a bijective mapping. Relations in a set. Equivalence relations. Partial and total order relations.

Integers.
Operations and their properties. Ordering. Principle of induction. Division. Highest common factor. Bezout identity. Prime numbers characterization. Euclidean algorithm. Prime factorization. Existence of infinite primes. Congruences. Residue classes and operations. The ring of the residue class. Cancellation law and invertible elements modulo n. Linear congruence equations. Chinese remainder theorem. Euler's function.

Groups
Binary operations. Monoids. Groups. Commutative monoids and groups. Residue classes. Plane transformations. Klein's group. Dihedral group. Cancellation law. Multiplication table.
Subgroups. Criterions. Intersection of subgroups. Subgroup generated by a subset.
Action of a group on a set. Trivial action. Transitive actions. Orbits and stabilizers. Orbit equation. Conjugation in a group: conjugacy classes. Centralizer of an element. Center of a finite p-group. Classification of groups of order a square of a prime. Right and left cosets of a subgroup. Index of a subgroup. Lagrange's theorem.
Symmetric group. Cycles. Disjoint cycles. Conjugacy classes in the symmetric group. Transposition. Parity of permutations. Alternating subgroup.
Powers in groups and monoids. Cyclic groups. Order of an element. Subgroups of a cyclic group and their reciprocal position. Number of elements of a given order in a cyclic group.
Normal subgroups. Quotient group. Quotient over the center of a group.
Group and monoid homomorphism. Kernel and image of a homomorphism. Direct and inverse images of subgroups and normal subgroups. Isomorphism theorems. Homomorphisms from cyclic groups. Endomorphisms of cyclic groups.
Product of subgroups. Product of normal subgroups. Internal and external direct product. Direct product of cyclic groups.

Type of didactic activities
Frontal lectures and guided exercises