Course notes ( they will be available at the beginnig of the course)
M. Suzuki, Group Theory I, Springer Verlag
M. Suzuki, Group Theory II, Springer-Verlag
D.J.S Robinson, A Course in the Theory of Groups, Springer-Verlag
Learning Objectives
Knowledge acquired: Basic ideas and techniques in abstract group theory, with emphasis on some aspects of the theory of finite groups which is interesting for the research.
Competence acquired: that concerning some parts of the theory of abstract groups that may have fruitful intersections with other areas of mathematics.
Skills acquired (at the end of the course): To understand a research article about finite groups.
Prerequisites
Courses to be used as requirements (required and/or recommended)
Algebra II , Algebra III and knowledges of Linear Algebra.
Courses required:
Courses recommended:
Teaching Methods
CFU: 9
Total hours of the course (including the time spent in attending lectures, seminars, private study, examinations, etc...): 225
Hours reserved to private study and other individual formative activities: 153
Hours for lectures: 72
Hours for laboratory: 0
Hours for laboratory-field/practice: 0
Seminars (hours): 0
Stages (hours): 0
Intermediate examinations (hours): 0
Further information
Attendance of lectures, practice and lab:
Not mandatory
Teaching tools:
Office hours:
It will be available at the beginning of the course.
Contact
P.zza Ghiberti 27, 50122 Firenze
Tel: 055 2755404
E-mail: luigi.serena@unifi.it
Type of Assessment
Oral exam
Course program
Fundamental concepts of Group Theory and first aspects of the Theory of permutation groups. Sylow Theorems. Examples of finite simple groups: the alternating group An (n>4) and PSL(2,F) (|F|>3).
Products: cartesian, direct,, semidirect, wreath and central.
Generation of groups: free groups, presentation and varieties.
Finitely generated abelian groups: characterization of their structure.
Basic properties of nilpotent groups. Finite nilpotent groups and the characterization of some types of finite p-groups.
Finite soluble groups: Hall subgroups and Hall systems.
Transfer and focal subgroup. Fusion of elements in a finite groups: conjugation families and the Alperin Fusion Theorem. Conditions for the existence of normal p-complements and for non simplicity of finite groups: Theorems of Burnside, Frobenius and Grün.