### Cohomology of Groups SS21

#### Schedule

 Lecture: Tuesdays 12:00-13:30 Room 48-562 / videos Jun.-Prof. Dr. Caroline Lassueur Thursdays 12:00-13:30 Room 48-562 / videos Jun.-Prof. Dr. Caroline Lassueur Exercises: Wednesdays 14:00-15:30 BigBlueButton Bernhard Böhmler Office hours: on appointment

• Start: 20th of April 2021
• Public holidays, no lecture:
• Ascension Day: 13th of May
• Corpus Christi: 3rd of June
• Exam dates: 3rd of September (3pm to 6pm), 23rd of September (afternoon), 20th of October (afternoon)

Video recordings: On Panopto!

The videos of the lectures will be uploaded in Panopto and made available through OpenOLAT Tuesdays/Thursdays by noon.

• 20.4.21: Slides of the Introduction
• Please register for the exercise classes in the URM system by Friday of the first week of the lecture time.
We will grant reading rights for the videos of the lecture to all students registered in the URM as soon as possible.
• Start: the 1st lecture takes place on Tuesday the 20th of April 2021, 12:00-13:00, over Zoom. We will quickly discuss the aims and the organisation of the lecture.
• Wednesday, 21st of April: there is no exercise session. The Exercises start in the 2nd week.

#### Exercises

Exercise Sheet 1. Due date: Monday, 26th April 2021, 6pm.
Exercise Sheet 2. Due date: 03.05.2021, 6pm
Exercise Sheet 3. Due date: 10.05.2021, 6pm
Exercise Sheet 4. Due date: 17.05.2021, 6pm
Exercise Sheet 4bis. Präsenzblatt
Exercise Sheet 5. Due date: 25.05.2021, 10am
Exercise Sheet 6. Due date: 31.05.2021, 6pm
Exercise Sheet 7. Due date: 07.06.2021, 6pm
Exercise Sheet 8. Due date: 14.06.2021, 6pm
Exercise Sheet 9. Due date: 21.06.2021, 6pm
Exercise Sheet 10. Due date: 28.06.2021, 6pm
Exercise Sheet 11. Due date: 05.07.2021, 6pm
Exercise Sheet 12. Due date: 12.07.2021, 6pm
Exercise Sheet 13. Due date: 19.07.2021, 6pm

The regular Exercise Classes will start in the 2nd week of the lecture period.

Due dates: please hand in your exercises in OpenOLAT by the date given on the sheets. Hand-written solutions are recommended. Please use a file format which is easy to mark. If possible avoid haevy files You are welcome, and encouraged, to work in groups of several students if wished.

#### References and lecture notes

Literature:
The lecture is based mainly on the following textbooks and articles:
• [Bro94] K. E. Brown, Cohomology of groups. See [MathSciNet]
• [CR81] C. Curtis and I. Reiner, Methods of representation theory. Vol. I. See [MathSciNet]
• [Eve91] L. Evens, The Cohomology of groups. See [MathSciNet]
• [LT17] C. Lassueur and J. Thévenaz, Universal $$p′$$-central extensions. See [MathSciNet]
• [Rot95] J. J. Rotman, An introduction to the theory of groups. Fourth edition. See [MathSciNet]
• [Rot10] J. J. Rotman, Advanced modern Algebra. See [MathSciNet]
• [Wei94] C. Weibel, An introduction to homological algebra. See [MathSciNet]

Lecture notes:
• Week 1: Background Material on Group Theory: Semi-direct Products
• Week 2: Background Material on Group Theory: Presentations of Groups
• Week 3: Background Material on Module Theory
• Week 4: Homological Algebra: Chain Complexes
• Week 5: Homological Algebra: Cochain Complexes / Projective Resolutions
• Week 6: Homological Algebra: Ext and Tor / Cohomology of Groups: Modules over the Group Algebra
• Week 7: Cohomology of Groups: (Co)homology of Groups, The Bar Resolution
• Week 8: Cohomology of Groups: Cocycles and Coboundaries / Low degree cohomology / Cohomology of Cyclic Groups / Group Extensions
• Week 9: $$H^1$$ and Extensions / $$H^2$$ and Extensions
• Week 10: Restriction/Transfer/Induction/Coinduction in Cohomology
• Week 11: The Theorems of Schur and Zassenhaus / Burnside's Transfer Theorem
• Week 12: The Schur Multiplier
• Week 13: Projective Representations and the Projective Lifting Property
• Week 14: Central extensions and Universality

#### Oral Exam

In principle one should be able to explain the content of the lecture notes.
• Definitions, statements of the theorems/propositions/lemmata should be known;
• you should be able to explain short proofs as well as the main ideas lines of the longer proofs;
• the Exercises mentionend in the lecture notes are important for the understanding of the theory and can be subject to questions;
• you should be able to give concrete examples/counter-examples to illustrate the results;
• there will be questions on concrete examples;
• last but not least: be ready to write down formally the concepts and results you are explaining.
Special wishes for exam dates: should be expressed to me by mid-June.

#### Contents of the Lecture

• Semidirect products of groups
• Presentations of groups
• Elementary module theory
• Homological algebra
• Ext and Tor functors
• Homology and cohomology of groups
• Cohomology and group extensions
• Classifications of finite groups of a given order
• The Schur multiplier and central extensions
• Projective representations