Required in stage
Optional in stage
First term
Second term
Both terms
Not organised
Next year
Alternating years
External
Prerequisites
Taught by
Language of instruction
Duration
Identical coursesGroup Theory (B-KUL-G0A85A)
This course is taught this academic year, but not next year.Aims
This course offers a deepening knowledge of the theory of discrete groups, in particular the nilpotent, solvable en polycyclic groups. Also an introduction to homological algebra is provided.
By introducing these concepts, properties and many examples, the students learn how to reason within the language of groups and homological algebra and their intuition gets stimulated.
Previous knowledge
The student needs a good kowledge of linear algebra, as treated for example in the course "G0N27A lineaire algebra" and of the theory of algebraic structures (groups, rings) as treated for example in "G0N88A Algebra I"
Is included in these courses of study
Activities
5 ects. Group Theory (B-KUL-G0A85a)
Content
Introduction to representation theory of finite groups:
- Definition and examples
- Irreducible representations and complete reducibility
- Lemma of Schur
- Characters: definition and properties
Nilpotent, Solvable and Polycyclic Groups
• Nilpotent groups:
- Upper and lower central sequence
- Definition of nilpotent group and examples
- Finite nilpotent groups
- Finitely generated (torsion free) nilpotent groupsSolvable groups
• Polycyclic and polycyclic-by-finite groups:
- Definition of poly-P and P-by-Q groups
- Nilpotent groups are polycyclic
- Hirsch length
- The max-condition
- Fitting subgroup
An introduction to homological algebra
• Homological algebra
- Exact sequences
- (Co)chain complexes and their (co)homology
- Ext and Tor
• Application: an introduction to the cohomology of groups
- Extn and the definition of the cohomology of groups.
- Fixed points and the zeroth cohomology group
- Semi-direct product and the first cohomology group
Example: the group of isometries of an Euclidian space
- Group extensions and the second cohomology group,
with an application to crystallographic groups.
Course material
Study cost: 1-10 euros (The information about the study costs as stated here gives an indication and only represents the costs for purchasing new materials. There might be some electronic or second-hand copies available as well. You can use LIMO to check whether the textbook is available in the library. Any potential printing costs and optional course material are not included in this price.)
Course notes are available via the students association
1 ects. Group Theory: Exercises (B-KUL-G0R67a)
Content
See G0A85a.
Course material
Study cost: Not applicable (The information about the study costs as stated here gives an indication and only represents the costs for purchasing new materials. There might be some electronic or second-hand copies available as well. You can use LIMO to check whether the textbook is available in the library. Any potential printing costs and optional course material are not included in this price.)
Course notes + Toledo
Evaluation
Evaluation: Group Theory (B-KUL-G2A85a)
Explanation
The exam is an open book exam.