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.
  • The first exam has happened. Answers are here.
  • The second exam has happened. Answers are here.
• 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.