B5521 - ARITMETICA E GRUPPI (M-Z)

Learning outcomes

By the end of the course, students will have gained familiarity with concepts from set theory, arithmetic and modular arithmetic, and group theory. They will be able to independently apply this knowledge to prove algebraic statements using rigorous language

Course contents

Sets and operations between sets. Relations (partial orderings, equivalence relations, etc.). Functions.

Quotient set, canonical projection, partitions, and equivalence relations.

Natural numbers, integers, and rational numbers. The principle of induction. Proofs by induction.
Congruences and operations on residue classes modulo n. Modular arithmetic. Divisibility rules.

Combinatorics of finite sets. Binomial coefficients. Inclusion-exclusion principle.

Cardinality of sets (overview). Countable sets. ℝ is uncountable. The power set P(X) does not have the same cardinality as X.

Euclidean division in ℕ and ℤ. Euclidean algorithm for computing the greatest common divisor. Bézout's identity. Primality and irreducibility in ℤ.

Algebraic properties of ℤ/nℤ. Solving linear congruences. Systems of linear congruences: the Chinese Remainder Theorem. Euler's totient function, Euler's theorem, Fermat's little theorem. Application: RSA cryptography.

Groups: definition and examples. Cyclic, dihedral, symmetric, and alternating groups. General linear group. Basic properties of finite groups. Order of elements and subgroups; Lagrange’s Theorem.

Conjugacy relation. Center of a group and centralizer of an element. Conjugacy in symmetric groups.

Groups and subgroups; left and right cosets. Group homomorphisms: image and kernel. Normal subgroups and quotient groups.

Isomorphisms and automorphisms. Inner automorphisms.

Group actions on sets. Cayley’s embedding. Orbits and stabilizers. Cauchy’s Theorem.

Direct and semidirect products. Structure of finite groups whose order has favorable arithmetic properties (is prime; is the product of two primes; is a power of a prime, etc.). Structure of finite abelian groups (overview).

Sylow's theorems.

Readings/Bibliography

The topics covered in the course are standard and can be found in almost every textbook for introductory university courses in arithmetic and algebra.

Two excellent books, with opposite pedagogical approaches, are:

• Herstein, "Topics in Algebra", J. Wiley & sons
• Artin, "Algebra", Pearson

Teaching methods

Chalk and blackboard. Lecture notes.

Weekly exercise sheets to be submitted and graded.

Assessment methods

 

Written and oral exams. Weekly exercise solutions will be taken into account.

Teaching tools

 

All course material will be uploaded on Virtuale

Office hours

See the website of Alessandro D'Andrea