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ADVANCED ALGEBRA B

Degree course: 
Corso di Second cycle degree in MATHEMATICS
Academic year when starting the degree: 
2020/2021
Year: 
1
Academic year in which the course will be held: 
2020/2021
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 algebraic structures and their properties: groups, rings, polynomials, fields. Knowledge of basic results in linear algebra and matrix calculus.

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 of main results in representation theory of finite groups.

EXPECTED LEARNING OUTCOMES

At the end of the course the student will be able to:
- compute character tables of known groups;
- complete the character table of a group given partial information
- read off group theoretical properties of a group from its character table

Introduction of representation and equivalent notions: group algebras and modules.

Decomposition of modules as sums of irreducible modules: Mashke's theorem.

Character of a representation: inner product of characters and ortogonality relations. Character tables.
Induced character and Frobenius' reciprocity.
Applications to finite groups: Burnside's theorem.

Introduction of representation and equivalent notions: group algebras and modules.

Decomposition of modules as sums of irreducible modules: Mashke's theorem.

Character of a representation: inner product of characters and ortogonality relations. Character tables.
Induced character and Frobenius' reciprocity.
Applications to finite groups: Burnside's theorem.

I. Isaacs, Character Theory of finite Groups, Dover

J. L. Alperin, R. Bell, Groups and Representations, Springer

G. James, M. Liebeck, Representations and Character of Groups, Cambridge University Press

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.
The exercises presented are often taken from past written examinations: they can be found, together with other selected exercise, on the web page of the course.

For simple and short questions, ask the teacher immediately before or after the class. Email teacher for longer questions.
For further detail go to the web page of the course.

Professors

MONTI VALERIO

Borrowers