Group Theory

A.Y. 2020/2021
Overall hours
Learning objectives
Aim of this course is to present topics and fundamental theorems concerning Group Theory
Expected learning outcomes
To read and understand topics in advanced Group Theory
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.
The program and the material for the course will not change.
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
To present basic ideas of group theory

1.Reviewing of some standard basic topics on group product ( direct and semidirect product) and on commutator subgroup and its properties. Isomorphism Theorems
2. Group actions Orbits, stabilizers. Sylow's Theorems and applications. Burnside's formula and permutation character. Induced action. Simplicity of some groups
3. Generators and Relations. Generators, Frattini subgroup (Schur's theorem).Finitely generated abelian groups. Free groups. Word problem.
4. Nilpotent and soluble groups. Central series and nilpotent groups, Fitting subgroup. p-nilpotent groups. Fixed point free automorphims and Frobenius Groups. Soluble groups; Carter subgroups. Schmidt-Iwasawa's Theorem.
Prerequisites for admission
Basics of Group Theory studied in Algebra 2
Teaching methods
Teaching Resources
A.Machì "Gruppi" Springer (2007)
-I.M.Isaacs " Algebra : a graduate course"Brooks/Cole Publishing Company(1993/4)
-B.A.F Wehrfritz "Finite groups" Word Scientific 1999
-D.J.Robinson " A course in the Theory of Groups" Springer-Verlag (1982)
Assessment methods and Criteria
The final examination consists of an oral exam.

- In place of a single oral exam, the student may choose instead to give a seminar immediatly after the conclusion of the course. The result will be available in the SIFA service through the UNIMIA portal.
- In the oral exam, the student will be required to illustrate results presented during the course and will be required to solve homeworks regarding Group Theory 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 three parts (written, oral, lab) 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
Lessons: 42 hours
Educational website(s)