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TOPICS IN ADVANCED ALGEBRA B

Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
2024/2025
Year: 
1
Academic year in which the course will be held: 
2024/2025
Course type: 
Compulsory subjects, characteristic of the class
Credits: 
8
Period: 
Second semester
Standard lectures hours: 
64
Detail of lecture’s hours: 
Lesson (64 hours)
Requirements: 

Knowledge of basic results in group theory: subgroups, homomorphisms, conjugation classes, Lagrange's theorem, isomorphism theorems, direct and semi-direct products, Sylow's theorem, classification of finitely generated abelian groups.

Final Examination: 
Orale

Written and oral exam.
The written examination lasts 2 hours and 30 minutes and tipically consists of 2 or 3 exercises divided in subquestions.
The oral examination starts immediately after the written part and usually begins with the discussion of the written test. The student will then be required to present some of the results seen in class. It will be object of evaluation the ability to present a proof in a complete and rigorous.

Passing examination and the final grading depend both on oral and written tests.

Assessment: 
Voto Finale

TEACHING OBJECTIVES
Knowledge od some topics in group theory: free groups and presentations of groups and nilpotent groups.
EXPECTED LEARNING OUTCOMES
At the end of the course the student will be able to study the properties of a group that can be derived from its presentation. Moreover he will use commutator calculus to analyze a group.

Congruences in monoids and groups. Relations and relators.

Free monoids and groups. Definition and elementary properties. Existence and unicity. Nielsen-Schreier theorem.

Presentations of groups. Tietze transformations. Todd-Coxeter method.

Nilpotent groups. Upper and lower central series. The normalizer condition. Finite nilpotent groups. The Hall collection formula. The Lie algebra of a nilpotent group.

Frontal lectures. Attending lectures is not mandatory, but strongly recommended.

Lectures are given at the board. Every topic is explained together with exercises useful to understand and apply exposed results. Sometimes the solution is given immediately, sometimes in a subsequent lecture in order to stimulate student to autonomous work. Especially at the end of the course, lectures of summarizing exercises are scheduled, in order to choose the more appropriate method to solve an exercise and to establish connections between different topics.

For simple and short questions, ask the teacher immediately before or after the class. Email teacher for longer questions (valerio.monti@uninsubria.it).
For further detail go to the web page of the course (algebrainsubria.altervista.org).

Bachelor students can borrow this course as Fundamentals of Advanced Algebra

Professors

MONTI VALERIO

Borrowers