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

Algebra is an essential component of mathematical literacy: it teaches students to recognise common structures in different problems and to move from concrete computations to general formulations. Group theory, which occupies the second part of the course, provides many students with their first systematic encounter with the kind of abstraction characteristic of modern mathematics. Groups formalise the notion of symmetry and have applications outside mathematics, including in physics, crystallography, chemistry and cryptography.

By the end of the course, students will be familiar with the basic language of sets, relations and functions; with the main tools of integer arithmetic and modular arithmetic; and with the fundamental notions of group theory, with particular emphasis on finite groups.

Students will be able to perform computations in these areas, recognise and construct examples, apply the main results discussed in the course, and independently address elementary algebraic problems and proofs.

The first part of the course introduces and organises some basic elements of mathematical language: sets and operations on sets, functions, equivalence relations and order relations, quotient sets, and the principle of mathematical induction. Basic elements of combinatorics and some introductory notions concerning the cardinality of sets will also be discussed.

The course will then cover integer arithmetic and congruences: Euclidean division, the Euclidean algorithm, Bézout’s identity, prime numbers, residue classes and modular arithmetic. The Chinese remainder theorem, Euler’s theorem and Fermat’s little theorem will be discussed, together with selected applications, including RSA cryptography.

The second part of the course will be devoted to group theory. The main examples, including cyclic groups, dihedral groups and permutation groups, will be introduced, followed by the study of subgroups, cosets, homomorphisms, isomorphisms, normal subgroups and quotient groups. Further topics will include Lagrange’s theorem, conjugation, group actions on sets, direct and semidirect products, and the main results concerning finite groups, including the theorems of Cauchy and Sylow. Some applications of these tools to the study of the structure of finite groups will also be presented.

Readings/Bibliography

Reference textbooks

I. N. Herstein, Algebra, Editori Riuniti.

M. Artin, Algebra, Bollati Boringhieri.

Lecture notes prepared by the instructor will also be distributed, together with more precise indications of the relevant sections of the textbooks.

Teaching methods

Chalk and blackboard.

Lecture notes will be distributed at irregular intervals. Exercise sheets will also be assigned during the course, generally on a weekly basis, and will be collected and assessed.

Assessment methods

The examination consists of a written test and an oral examination, both of which are compulsory.

The written test is intended to assess the student’s ability to use the tools and results presented during the course to solve exercises and problems. Students who obtain a mark of at least 18/30 in the written test are admitted to the oral examination.

The oral examination assesses the student’s knowledge and understanding of the topics covered in the course, as well as their ability to present definitions, examples, results and proofs. The final grade takes into account the outcome of both parts of the examination.

Completion of the assignments given during the course may earn students a bonus of up to five points. The bonus is added exclusively to the numerical result of the written test at the first examination session and cannot be used in subsequent examination sessions.

During the written test, students may not use books, notes, calculators, mobile phones or other electronic devices. Communication with other people and the use of artificial intelligence tools are not permitted.

Teaching tools

Lecture notes, exercise sheets and other teaching materials will be made available through the Virtuale platform.

Office hours

See the website of Alessandro D'Andrea