### Cohomology of Groups SS18

#### Schedule

 Lecture: Tuesdays 11:45-13:15 Room 48-438 Lecturer: Caroline Wednesdays 13:45-15:15 Room 48-438 Lassueur Exercises: Wednesdays 10:00 - 11:30 Room 48-582 Assistant: Patrick Wegener Office hours: Mondays 11:00-12:00, Room 48-409 14:00-15:00

Exam Dates:
• 24.07.2018
• 08.10.2018
• 22.10.2018

#### Updates and Information

• Please register for the exercises in the URM system by Friday, 13th of April 2018, noon.
• The first two weeks of the lecture will be given by Patrick Wegener. He will use this opportunity to go through some background material in group theory.
• Here are the slides of the Introduction.
• Tuesday, 1st of May is a public holiday: there is no lecture.
• On Wednesday, 23rd of May the lecture is replaced by a tutorial. Optional Exercises
• On Wednesday, 11th of July, the lecture room is occupied by the IRTG-Seminar. On this day, the lecture will take place from 10:00 to 11:30 in 48-582, and the Exercises from 13:45 to 15:15 in 48-436.
• 12th of July: some tips to prepare the oral exam can be found below.
• 16th of July: The lecture notes have been updated with typos and suggestions mentionned to me corrected.

#### Exercises

Registration: Please register for the exercises in the URM system by Friday, the 13th of April 2018, noon.

The exercise sheets will be uploaded on the handing-in day of the previous one.

Exercise Sheet 1. Due date: 17.04.2018, 10:00
Exercise Sheet 2. Due date: 24.04.2018, 10:00
Exercise Sheet 3. Due date: ⚠ 30.04.2018, 18:00
Exercise Sheet 4. Due date: 08.05.2018, 10:00
Exercise Sheet 5. Due date: 15.05.2018, 10:00
Exercise Sheet 6. Due date: 22.05.2018, 10:00
Exercise Sheet 7. Due date: 29.05.2018, 10:00
Exercise Sheet 8. Due date: 05.06.2018, 10:00
Exercise Sheet 9. Due date: 12.06.2018, 10:00 (⚠ there was a correction to Ex. 1)
Exercise Sheet 10. Due date: 19.06.2018, 10:00
Exercise Sheet 11. Due date: 26.06.2018, 10:00
Exercise Sheet 12. Due date: 03.07.2018, 10:00

The regular Exercise Classes will start in the 2nd week. On Wednesday, 11th of April, there will be a short facultive tutorial reviewing examples of groups we will use throughout the lecture.

Due dates: please hand in your exercises in the dedicated letter-box next to Lecture Room 48-208. You are welcome 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:
I will write lecture notes for this lecture. As I take it for the first time, the lecture notes will progressively be available in the course of the semester.

Please, let me know if you find typos (of all kinds). I will correct them. Comments and suggestions are also welcome.

#### 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 red lines of the longer proofs;
• the Exercises mentionend in the lecture notes are important for the understanding of the theory;
• you should also be able to give concrete examples/counter-examples to illustrate the results;
• there will also be questions on concrete examples;
• there won't be direct questions on Chapter 2 and the Appendix. There won't be questions on the last lecture either.
• Please, be ready to write down formally the concepts and results you are explaining.

#### Contents of the Lecture

• Semidirect products of groups
• Presentations of groups
• Homological algebra
• Homology and cohomology of groups
• Cohomology and group extensions
• The Schur multiplier and central extensions
• Projective representations

Prerequisites: elementary group theory and linear algebra

#### Workload

Lecture : 4 SWS, i.e. 60 contact hours
Exercises : 2 SWS, i.e. 30 contact hours
Personal work : 210h (recommended)
Credit points : 9