Algebra 2

A.Y. 2021/2022
Overall hours
Learning objectives
Aim of this course is the introduction of the principal properties of some algebraic structures: semigroups and groups.
Expected learning outcomes
Students will acquire basic knowledge regarding some fundamental algebraic structures (semigroups and grups) and will be able to develop elementary proofs of properties concerning groups.
Course syllabus and organization

Single session

Lesson period
First semester
- Both lectures and exercise classes will be delivered through the application Zoom, and there will be the possibility to recover them at any time because they will be recorded and posted on ARIEL. There will be also "laboratory" activities, in small groups, to be carried out in presence or through Zoom according to the requests of the various groups of students, to resume and deepen topics of the course, and to facilitate the relationship between students.

- The program and the material for the course will not change.

- Both the written part and the oral part of the exam will take place through the Zoom application until it will be necessary to do so; there are no substantial changes on the examination methods, only logistical issues that will be communicated in due course via the ARIEL platform.
Course syllabus
We introduce the algebraic structures in particular groups.
Groups and their fundamental properties. Subgroups and cosets. Group homomorphisms.
Normal subgroups and factor groups. Cyclic groups, linear groups,
permutation groups. Lagrange's Theorem, commutators and commutator subgroup. Direct products. Group actions: stabilizers, orbits, transitivity, regularity, Cayley's Theorem. p-groups and Sylow's Theorem.
Endomorfisms of cyclic groups and automorfisms of cyclic groups. Preliminaries on solvable groups.
Prerequisites for admission
Basics of Algebra studied in Algebra 1
Teaching methods
Traditional Lectures
Teaching Resources
D.Dikranjan, M.S.Lucido Aritmetica e Algebra, Liguori Editore
M. Isaacs "Algebra, a graduate course" Brooks /Cole Publishing Company
Assessment methods and Criteria
The final examination consists of two parts: a written exam, an oral exam.

- During the written exam, the student must solve some exercises in the format of open-ended , with the aim of assessing the student's ability to solve problems similar to those done during the course . The duration of the written exam will be proportional to the number of exercises assigned, also taking into account the nature and complexity of the exercises themselves (however, the duration will not exceed three hours). In place of a single written exam given during the first examination session, the student may choose instead to take 2 midterm exams. The outcomes of these tests will be available in the SIFA service through the UNIMIA portal.
- The oral exam can be taken only if the written component has been successfully passed. In the oral exam, the student will be required to illustrate results presented during the course in order to evaluate her/his knowledge and comprehension of the arguments covered as well as the capacity to apply them.

The complete final examination is passed if all two parts (written, oral) are successfully passed. Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.
MAT/02 - ALGEBRA - University credits: 6
Practicals: 36 hours
Lessons: 27 hours
Educational website(s)