B5521 - ARITMETICA E GRUPPI (M-Z)

Learning outcomes

Successful students will be familiar with basic notions of set theory, arithmetics and (mostly finite) group theory.

They should be able to apply their knowledge to state and prove algebraic statements in a formally rigorous language.

Course contents

Operations between sets. Relations: partial and total orderings; equivalence relations. Equivalence classes and quotient sets; partitions.

Injective, surjective, biujective maps. Composition and inverse map.

Peano axioms for natural numbers. Induction and proof by induction. Construction of integer, rational, real, complex numbers. Fundamental theorem of algebra.

Modular arithmetic: ring structure on Z/n. Divisibility criteria.

Prime and irreducible (integer) numbers. Primes are irreducible. Euclidean division in Z. GCD and lcm. Bèzout identity. Irreducibles are prime. Z/n is a field iff n is prime.

Chinese Remainder Theorem. Linear congruences. RSA method in cryptography.

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Groups: finite, cyclic, commutative. Examples. Order of groups and elements. Lagrange theorem.Symmetric and alternating groups. Cyclic notation for permutations.

Subgroups and their properties. Normal subgroups.

Homomorphisms and homomorphism theorems. Group actions on sets and Cayley's theorem. Isomorphisms and automorphisms.

Conjugacy classes and applications. Groups of order p^n. Sylow subgroups and Sylow theorems. Classification of finite groups of small order.

 

 

Readings/Bibliography

 

Every textbook for a first algebra university class will do.

Teaching methods

Frontal lectures on a blackboard.

Weekly problem sets.

Office hours.

Assessment methods

Written and oral exams.

Teaching tools

 

Correction of problem sets by tutors.

Office hours

See the website of Alessandro D'Andrea

See the website of Jacopo Gandini