Group Theory

Class overview

This lecture course will provide Master students in Math with some basic concepts of group theory and applications.

Organization:

+ 3 hours/ week (including exercise sessions)

+ Time of Lecture Course: 8:00-11:30  (Monday, Wednesday, Friday)

+ Exercise Class: 8:00-11:30  (Monday, Wednesday, Friday)

+ Room: Building L, Third Floor, Seminar Room 2 (Faculty of Math)

+ Written final examination (90-120 minutes)

Literature

[1] J.J. Rotman, An Introduction to the Theory of Groups, Fourth Edition, Springer-Verlag, 1994.

[2] A. Ayache, K. Amin, Introduction to Group Theory, Springer, 2025.

[3] H.E. Rose, A Course on Finite Groups, Springer, 2009.

Contents of the lecture course

Chapter I: Basic Concepts in Group Theory 

1.  Basic Definitions and Examples

2.  Subgroups, Cosets and Lagrange’s Theorem

3.  Normal Subgroups

Chapter II: Group Construction and Representation

4.  Permutation Groups

5.  Matrix Groups

6.  Group Presentation

Chapter III: Group Homomorphisms

7.  Homomorphisms and Isomorphisms

8.  Isomorphism Theorems

9.  Cyclic Groups - Automorphism Groups

Chapter IV: Finite p-Groups and Sylow Theory 

10.  Action and the Orbit–Stabiliser Theorem

11.  Finite p-Groups 

12.  Sylow Theory and Applications

Chapter V: Several Important Groups 

13.  Direct Products 

14.  Finite Abelian Groups

15.  Composition Series and the Jordan–Hölder Theorem

16.   Nilpotent Groups, Frattini and Fitting Subgroups

 Course Materials

• Lecture 1 and Exercises

• Lecture 2 and Exercises

• Lecture 3 and Exercises

• Lecture 4 and Exercises

• Lecture 5 and Exercises

• Lecture 6 and Exercises

• Lecture 7 and Exercises

• Lecture 8 and Exercises

• Lecture 9 and Exercises

• Lecture 10 and Exercises

Some Further Notes