# Group TheorySS 2020

Dietrich Burde

Lectures: Monday 13:15 - 14:00 in SR11 and Thursday 13:15 - 14:45 in SR11

Exercises: Thursday 15:00-15:45 in SR10

• This page contains informations and pdf-files for this lecture and its exercise class.
• Group theory is a broad subject which arises in many areas of mathematics and physics, and has several different roots. One foundational root of group theory was the quest of solutions of polynomial equations of degree higher than 4. Lagrange introduced permutation groups for the theory of equations, and Galois the groups named after him for the solvability of the equation with radicals. A second root was the study of symmetry groups in geometry. The systematic use of groups in geometry was initiated by Klein's 1872 Erlangen program. Finally, a third root of group theory was number theory. Certain abelian group structures had been implicitly used in number-theoretical work by Gauss, and more explicitly by Kronecker.
Modern group theory nowadays is not just a part of abstract algebra. It has several branches, such as combinatorial group theory, geometric group theory, the theory of finite groups, the theory of discrete groups, transformation groups, Lie groups and algebraic groups, and many more. This lecture covers the topics stated in the curriculum for master in mathematics at the university of Vienna.

• Here is a syllabus and a bibliography available.

• Exercises: Participation is only possible for those present in the first meeting at the beginning of March. In general, presence is mandatory for the exercise class. Students being absent more than one time will be signed out. This will not apply in case of illness with a medical certificate. For details please ask at the first meeting. There are 42 exercises in total. Exercises 7,14,21,28,35,42 are extra tasks, which are voluntary. For each week everyone is supposed to prepare three exercises, i.e., we start with 1,2,3 for the second week, then 4,5,6 (and extra 7 if you like), etc. There might be a short test, too.

Exam 2020:

See here.

## pdf-files:

 No. Topic Date pdf-file 1 Lecture notes 2020 group_theory.pdf 2 Exercises 2020 exercises.pdf

Topics for the exam:

1. Basic notions and examples, Sylow theorems and semidirect products.
2. Subnormal series, solvable and nilpotent groups
3. Free groups and presentations by generators and relations
4. Group extensions and factor systems
5. Cohomology of groups

Dietrich Burde