### Character Theory of Finite Groups SS 2023

#### Schedule

Lecture: |
Mondays | 10:00 - 11:30 | Room 48-538 | Lecturer: |
Jun.-Prof. Dr. Caroline Lassueur |

Exercises: |
Fridays | 11:45 - 13:15 | Room 48-438 | Instructor: |
Annika Bartelt |

Office hour: |
Upon request |

**Link to the live stream**: From Week 2 on in OpenOLAT over BBB.

**Public holidays / Exceptional schedule:**

- Friday, the 28th of April: the 1st Exercise Class is replaced by a lecture.
- 1st of May 2023: Labour Day - no lecture
- 29th of May 2023: Pentecost Monday - no lecture

**Office hours**: you can make appointments with me for individual qusetion sessions.

**Exam Dates:**

- With U. Thiel: 31.07.2023
- With A. Gathmann: 29.08.2023

#### Updates

- April 2023:
**⚠**please register in the URM system by Friday, the 24th of April, midday.

#### Lecture Notes

I will essentially follow my

You can use the version of the SS 2022 for Chapters 1 to 6.

**\(LaTeX\)ed lecture notes from the SS 2022**, up to small alterations.You can use the version of the SS 2022 for Chapters 1 to 6.

**⚠**In the SS 2023 I will add a Chapter 7 on Frobenius Groups, which will be treated in the last lectures. Here are the**updated full \(LaTeX\)ed lecture notes of the SS 2023**including Chapter 7.**Here are the topics treated in the corresponding weeks (updated at the end of the lecture):****Weeks 1 & 2**:**Chapter 1**: Linear Representations of Finite Groups +**Appendices A and B**: Modules and Algebras [slides.pdf]**Week 3**: no lecture, public holiday**Week 4**:**Chapter 2**: §4 Modules over the group algebra & §5 Schur's Lemma and Schur's relations**Week 5**:**Chapter 2 + Chapter 3**: §6 Representations of finite abelian groups & §7 Characters**Week 6**:**Chapter 3**: §8 Orthogonality of characters, §9 Consequences of the 1st Orthogonality Relations**Week 7**: no lecture, public holiday**Week 8**:**Chapter 3 + Chapter 4**: §10 The regular character, §11 The character table of a finite group, §12 The 2nd Orthogonality Relations, §13 Tensor products of represenstions and characters,

Slides_X(Cn)_X(S3).pdf, Appendix C: tensor products of vector spaces**Week 9**:**Chapter 4 + Chapter 5**: §14 Normal subgroups and inflation, §15 Algebraic integers and character values

Slides_X(S4).pdf, Appendix D: Integrality and algebraic integers**Week 10**:**Chapter 5**: §16. Central characters, §17 The centre of a character

Slides_DegreesIrr(GL_3(2)).pdf**Week 11**:**Chapter5 + Chapter 6**: §18 Burnside's \(p^aq^b\) Theorem, §19 Induction and restriction**Week 12**:**Chapter 6**: §19 Induction and restriction, §20 Clifford Theory

Slides_X(A5).pdf**Week 13**:**Chapter 6**: §20 Clifford theory, §21 The theorem of Gallagher**Week 14**:**Chapter 7**: Frobenius Groups

#### Exercises

The Exercise Classes take place fortnightly. The Exercise Sheets will be uploaded below.

Please hand in your solution sheets on Wednesdays, at the latest at 6 pm in the

**Friday, the 28th of April**: the 1st Exercise Class is replaced by a lecture, in order to make up for the two public holidays during the 1st half of the semester. This way the exercises will let us cover the material seen in the lectures in a more consistent way.**Exercise Sheet 1**. Due date: 10.05.2022, 4pm**Exercise Sheet 2**. Due date: 24.05.2023, 4pm**Exercise Sheet 3**. Due date: 07.06.2023, 4pm**Exercise Sheet 4**. Due date: 24.06.2023, 4pm**Exercise Sheet 5**. Due date: 05.07.2023, 4pm**Exercise Sheet 6**. Due date: 19.07.2023, 4pm

**Handing in solutions**:Please hand in your solution sheets on Wednesdays, at the latest at 6 pm in the

**dedicated letter-box next to Lecture Theater 208**or upload them in OpenOLAT.- You should hand in your solutions in handwritten form by the due date. No LaTeXed solutions accepted.
- You can hand in in groups of up to two students.

#### Übungsscheine

You obtain an "Übungsschein" if the following criteria are fulfilled:

- you have obtained at least 40% of the possible points on each exercise sheet;

(handing-in in groups of two students is possible) - you have actively taken part to the exercise classes: attendance to the exercise classes + presenting at least two solutions on the board during the semester.

#### References

**Textbooks you can use are the following. However, the lectures do not follow any of them faithfully.**

- [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.

#### Oral Exam

In principle one should be able to explain the content of the lecture.
Warning: this semester I have added a Chapter 7 about Frobenius groups to the lecture notes.

A typical exam question would be as follows: Explain all entries in the character table on the cover page picture and find the missing entries. (Several methods possible!)

- 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.

A typical exam question would be as follows: Explain all entries in the character table on the cover page picture and find the missing entries. (Several methods possible!)