Algebra 2
A.Y. 2024/2025
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.
Lesson period: First semester
Single course
This course can be attended as a single course.
Course syllabus and organization
Single session
Responsible
Lesson period
First semester
Course syllabus
We introduce the algebraic structures in particular groups.
Modules over a ring, modules over a PID, structure theorem with applications. 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 and automorfisms of groups. Preliminaries on solvable groups. Finitely generated abelian groups.
Modules over a ring, modules over a PID, structure theorem with applications. 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 and automorfisms of groups. Preliminaries on solvable groups. Finitely generated abelian groups.
Prerequisites for admission
Basics of Algebra studied in Algebra 1
Teaching methods
Classical lectures
Teaching Resources
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.
- 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
Lessons: 27 hours
Professors:
Binda Federico, Montoli Andrea
Shifts:
Educational website(s)
Professor(s)
Reception:
By appointment only, on Thursday 10:30-12:30
Office 2093
Reception:
by appointment via e-mail
office 1014, Via Saldini 50