# Topics in Group Theory

This website is for the 4th year course at the Universidad de Costa Rica starting Monday 10th March, 2014.

**Practical matters**

**Lectures**: There are two lectures per week (Monday, 15:00 – 17:50 and Thursday, 16:00 – 17:50) in Room 400. There will be 80 hours of lectures in total.**Lecture notes**: These will be put online on Thursday afternoon each week.**Office hours**: Mondays 2 – 3pm and Thursdays 2 – 4pm.**Assessment**: There will be two exams, each counting 40% of the total mark. The remaining 20% of the mark will be for exercises.**La carta al estudiante**(in Spanish).

**Lecture notes**

- Lectures 1 and 2 – Introduction and Category Theory;
- Lecture 3 – Group actions.
- Lecture 4 – The alternating groups.
- Lecture 5 – Primitive permutation groups.
- Lecture 6 – The product action.
- Lecture 7 – Minimal normal subgroups and the socle.
- Lecture 8 – The O’Nan-Scott Theorem.
- Lecture 9 – Series.
- Lecture 10 – Fields and vector spaces.
- Lecture 11 – Projective space.
- Lecture 12 – Linear groups acting on projective space.
- Lecture 13 – Forms and polar spaces.
- Lecture 14 – Isometries and Witt’s lemma.
- Lecture 15 – Abstract polar spaces. (This lecture is in draft form; its content is not officially part of the course.)
- Lecture 16 – Symplectic groups.
- Lecture 17 – Unitary groups.
- Lecture 18 – Orthogonal groups.

**Exercises**

I will provide full answers for the first set, thereafter answers will only be provided on request.

- Exercises 1 and 2 – Answers 1 and 2.
- Exercises 3 – Answers 3.
- Exercises 4.
- Exercises 5.
- Exercises 6.
- Exercises 7.
- Exercises 8.
- Exercises 9.
- Exercises 10.
- Exercises 11.
- Exercises 12.
- Exercises 13.
- Exercises 14.
- Exercises 16.
- Exercises 17 and 18.

**Background reading**

No one text covers all of the material in this course. Principal texts are as follows:

- John D. Dixon and Brian Mortimer,
*Permutation groups*, Graduate Texts in Mathematics, vol. 163, Springer-Verlag, New York, 1996. - Peter Cameron’s notes on Classical Groups;
- Peter Cameron’s notes on Projective and polar spaces (First edition published as QMW Maths Notes 13 in 1991).

Additional texts of interest:

- Harald Simmons, An introduction to category theory. (I drew on a very small part of these notes when I wrote the chapter on category theory.)
- Jean Dieudonne,
*La geometrie des groupes classiques*. This is a classic, in French. - Peter Kleidman and Martin Liebeck,
*The subgroup structure of the finite classical groups*. This proves a refined version of Aschbacher’s theorem on the subgroup structure of the finite classical groups. It also contains a wealth of other information on these groups (and other almost simple groups). - Donald Taylor,
*The geometry of the classical groups*. This covers all the material in the second half of this course plus a fair bit more. - Helmut Wielandt,
*Finite permutation groups*. Another classic giving a good sense of the major themes in the development of the theory of finite permutation groups. - Robert Wilson,
*Finite simple groups*. Related lecture notes are online. - Joanna Fawcett, The O’Nan-Scott theorem for finite primitive permutation groups. A very nice Masters thesis giving a self-contained proof of the O’Nan-Scott theorem.

I have e-copies of most of the texts listed above and can provide them on request.