### Character Theory of Finite Groups SS 2020

#### Schedule

 Lecture: Tuesdays 10:00 - 11:30 Room 48-438 Lecturer: Jun.-Prof. Dr. Caroline Lassueur Exercises: Fridays 11:45 - 13:15 Room 48-438 Assistant: Birte Johansson M.Sc. Office hour: See OpenOLAT

Office hours: See OpenOLAT for online office hours. You can also make appointments with me for individual online qusetion sessions.

Exam Dates: (with W. Hart und D. Pol)
• 15. Sept. 2020 (nachmittags)
• 23. Okt. 2020 (nachmittags)
Registration must be done with Frau Sternike: by E-Mail.

• 14th April: Introductory slides
• 14th April: link to OpenOLAT
• IMPORTANT: If you are interested in attending the lecture, please register in the URM system, so that we can easily contact you by e-mail.

#### Lecture Notes

$$LaTeX$$ed lecture notes will be uploaded here every Tuesday before 10am.

Lecture Notes:
• Foreword
• Conventions
• Notation index
• Weeks 1+2: Chapter 1: Linear representations of finite groups [Explanation]
• Week 3: Appendices A and B: Modules and Algebras and
§4 Modules over the Group Algebra [Explanation]
• Week 4: §5 Schur's Lemma, §6 Representations of Finite Abelian Groups [Explanation]
• Week 5: §7 Characters, §8 Orthogonality of Characters (part 1) [Explanation]
• Week 6: §8 Orthogonality of Characters (part 2), §9 Consequences of the 1st Orthogoinality Relations, §10 The regular character [Explanation]
• Week 7: §11 The Character Table of a Finite Group, §12 The 2nd Orthogoinality Relations, Appendix C Tensor Products of Vector Spaces [Explanation]
• Week 8: §13 Tensor Products of Representations and Characters, §14 Normal Subgroups and Inflation [Explanation]
• Week 9: §14 Normal Subgroups and Inflation (part 2), §15 Algebraic Integers and Character Values and Appendix D: Integrality and Algebraic Integers [Explanation]
• Week 10: §16 Central Characters, §17 The Centre of a Character [Explanation]
• Week 11: §17 The Centre of a Character (continued), §18 Burnside's $$p^aq$$ Theorem [Explanation]
• Week 12: §19 Induction and Restriction [Explanation]
• Weeks 13+14: §20 Clifford Theory, §21 Theorem of Gallagher [Explanation]
• Full text
In OpenOLAT you will also find sum-ups of each lecture. (Uploaded each Tuesday.)

#### Exercises

The Exercises take place fortnightly.
Registration: Please register in the URM system by Friday, the 17th of April 2020, noon.

The exercises begin in the 2nd week of the lecture period and then take place every second week. From Sheet 2 on, the exercise sheets will be uploaded on the hand-in day of the previous one.

Exercise Sheet 1. Due date: 23.04.2020, 13:00 in OpenOLAT
Exercise Sheet 2. Due date: 07.05.2020, 13:00 in OpenOLAT
Exercise Sheet 3. Due date: WEDNESDAY 20.05.2020, 13:00 in OpenOLAT
Exercise Sheet 4. Due date: 04.06.2020, 13:00 in OpenOLAT
Exercise Sheet 5. Due date: 18.06.2020, 13:00 in OpenOLAT
Exercise Sheet 6. Due date: 02.07.2020, 13:00 in OpenOLAT
Exercise Sheet 7. The last exercise sheet need not be handed in.

Handing in solutions: you should hand in your solutions in OpenOLAT by the due date.

#### Übungsscheine

You obtain an "Übungsschein" if the following criteria are fulfilled:
• you have obtained at least 50% of the points on the Exercise Sheets (1 to 6) altogether.
• you have obtained at least 1 point in 2 Exercises in each Exercise Sheet (1 to 6)

Each exercise is worth 4 points.

Exercises can be handed in groups of two students, if you wish. However, as long as the lecture takes place online, we strongly recommend that everyone tries to hand in all exercises on their own, as it will allow you. to get a deeper understanding of the topic and make sure you are up to date.

#### References

Textbooks:
• [JL01] G. James and M. Liebeck, Representations and characters of groups. See [zbMATH].
• [Ser77] J.-P. Serre, Linear representations of finite groups. See [zbMATH].
The original text is:
[Ser98] J.-P. Serre, Représentations linéaires des groupes finis. See [zbMATH].
• [Isa06] M. Isaacs, Character theory of finite groups. See [zbMATH].
• [Web16] P. Webb, A course in finite group representation theory. See [zbMATH].
• [CCNPW85] J.H. Conway, R.T. Curtis, S.P. Norton, R. Parker, R.A. Wilson, Atlas of Finite Groups. Clarendon Press, Oxford, 1985.
As a complement to my lecture notes, I strongly recommend Peter Webb's book, whose pre-print version is available at his webpage.

#### Oral Exam

In principle one should be able to explain the content of the lecture.
• Definitions, statements of the theorems/propositions/lemmata should be known.
• You should be able to explain short proofs as well as the main arguments of the longer proofs.
• The Exercises mentioned in the lecture are important for the understanding of the theory.
• There won't be any direct questions on the content of the Appendices.
• you should also be able to give concrete examples/counter-examples to illustrate the results.
• There will also be questions on concrete examples.
• Also be ready to write down formally the concepts and results you are explaining.
In particular you should be able to expalin in your own words everything that is contained in the sum-ups uploaded in OpenOLAT.

#### Contents of the Lecture

• Linear representations and characters
• Modules over the group algebra
• Character tables, orthogonality relations
• Burnside's $$p^aq^b$$-theorem
• Restriction, induction, inflation, tensor products
• Clifford theory
• Frobenius groups (if time permits)

Prerequisites: elementary group theory and linear algebra