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


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

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.

Background literature on growth and expanders

The material in here is fairly comprehensive. Comments in red describe the sources that I suggest you start with. Back to main seminars page.

Background on expanders:

On the sum-product phenomenon. The basic text is Tao and Vu “Additive Combinatorics”. Here are a few other links:

  • Green, Sum-product over F_p. This is a survey article.
  • Green, Sum-product over the complexes. This focusses on Solymosi’s result.
  • Bourgain, Katz, Tao, Sum-product over F_p. This is the first proof of sum-product over F_p. However it is difficult so I wouldn’t recommend it as a first read. Plus it is not completely general (it doesn’t work for very small sets). On the plus side it very clearly demonstrates the connection between sum-product phenomena and Szemeredi-Trotter theorems. So it’s worth looking at for this reason.
  • Helfgott, Growth in SL_3. Section 3 of this paper is devoted to a generalization of the idea of the sum-product theorem. It is based on the Glibichuk-Konyagin sum-product theorem over F_p (which works for small sets so improves the work of BKT mentioned above), but is couched in much more general terms; the statement involves one group acting upon another. This is the key statement of the sum-product phenomenon for our purposes; the exposition is excellent.
  • Tao and Vu, Additive Combinatorics. To get a taste of how incidence theorems are related to the sum-product phenomenon read Sections 8.1-8.3.

On growth in non-abelian groups:

  • Helfgott, Growth in SL_3. This paper is central. It includes a discussion of key ideas including escape, non-singularity, sticking subgroups in different directions, bounds on torus intersection, bounds on conjugacy classes, use of incidence theorems etc. The statement of Theorem 1.1 is “ideal” (as this is currently understood); it reduces questions of growth to the nilpotent setting.

    Sections 4 and 5 of this paper are particularly vital. Subsequent sections of the paper use incidence theorems; we will be able to go to the full result more directly.

  • Breuillard, Green, Tao, Growth in Chevalley groups. This is an announcement of the general result which focusses on the pivot argument. A full proof of this result does not yet exist in the literature. So for now the best exposition of the result is this blog entry of Tao (see also the subsequent discussion, especially the simplifications described by Helfgott).
  • Pyber, Szabo, Growth in groups of Lie type. The methods in this paper are similar to those of the previous link. However the statement is more general (it holds for twisted groups). In addition examples are given that justify the dependency of the constant epsilon on the rank.
  • Larsen, Pink, Subgroups of algebraic groups. This paper contains material that was central to adapting Helfgott’s work to the general setting (a la the previous two points).

Expanders from groups:

  • Bourgain, Gamburd, Expanders in SL_2(p). This uses Helfgott’s result on growth in SL_2(p) to generate expanders. These two clever people have a couple of other papers that are also of interest:
    • J. Bourgain and A. Gamburd, Expansion and random walks in
      SLd (Z/pn Z):I, J. Eur. Math. Soc. 10 (2008), 987-1011.

    • J. Bourgain and A. Gamburd,
      SLd (Z/pn Z):II, preprint
  • Varju, Expansion in SL_d This generalizes some of the work of Bourgain and Gamburd.
  • Kassabov, Nikolov, Lubotzky, Finite simple groups as expanders. An entirely different set of techniques.
  • Kassabov, Symmetric groups as expanders. Another entirely different set of techniques.


Property T: The first construction of expander graphs was by Margulis and used property T, a representation theoretic property that holds for certain discrete groups (SL_d(Z) with d>2 for instance).

  • Wikipedia gives a definition of Property T.
  • A. Lubotzky. Discrete groups, expanding graphs and invariant measures, volume 125 of Progress in Mathematics. This book outlines (amongst other things) the connection between Property T and expansion properties.

Galois Theory

Harald Helfgott and I are teaching Galois Theory in Teaching Block 1, starting October 2009. Information about (my part of) the course will appear below.

  • Course page: A description of the course, as it appeared in the handbook, can be found here
  • Timetable: Lectures are at 9am on Monday, Wednesday, and Friday, starting Monday 5th October. The venue will be Portacabin Room 1. Great!
  • Notes: Click here for notes covering the first nine sections (up to Normal Extensions).
  • Information about assessment: Exercise Sheets 1, 2, and 3 are available. I am not going to write up solutions unless I get a specific request (everything will be covered in the exercise class). Exam information will appear here in due course.
  • Texts: I recommend you refer to the book by Ian Stewart, or the one by D.J. Garling. For Stewart’s book (editions 2 and 3), you can find errata, and other supplementary material, here.

Expanders and growth in groups

These are the web pages for a set of seminars to be given at the University of Western Australia in 2010. The seminars constitute an introduction to the area of “growth in groups”, and the connection between growth and “expander graphs”.

The research group at UWA is very strong in group theory and in finite geometry, hence I will emphasise these aspects of the subject. I will also assume familiarity with results from these areas.

  • Recommended background reading can be found here.
  • Lectures (I will add lecture notes after the lecture is given):
    1. Sum-product (12 noon, Tues 4 May 2010; notes now available)

      We introduce the idea of growth in groups, before focussing on the abelian
      setting. We take a first look at the sum-product principle, with a brief
      foray into the connection between sum-product results and incidence

      We then focus on Helfgott’s restatement of the sum-product principle in
      terms of groups acting on groups.

    2. Growth in groups of Lie type (11am, Fri 7 May 2010; notes now available)

      Since Helfgott first proved that “generating sets grow” in SL_2(p) and
      SL_3(p), our understanding of how to prove such results has developed a
      great deal. It is now possible to prove that generating sets grow in any
      finite group of Lie type; what is more the most recent proofs are very
      direct – they have no recourse to the incidence theorems of Helfgott’s
      original approach.

      We give an overview of this new approach, which has come to be known as a
      “pivotting argument”. There are five parts to this approach, and we
      outline how these fit together.

    3. Escape (11am, Fri 14 May 2010); notes now available)

      The principle of “escape from subvarieties” is the first step in proving
      growth in groups of Lie type. We give a proof of this result, and its most
      important application (for us) – the construction of regular semisimple

      We then examine other related ideas from algebraic geometry, in particular
      the idea of non-singularity.

    4. Growth in soluble subgroups of GL_r(p) (11am, Wed 21 Jul 2010; notes will not be written up for this – a preprint will appear on the arXiv in due course.)

      We show how to reduce the study of exponential growth in
      soluble subgroups of GL_r(p) to the nilpotent setting. We make use of ideas based on the sum-product phenomenon, as well as some machinery from linear algebraic groups. We will not assume any background from these areas. This lecture is based on new results of the lecturer and Helfgott.

    5. An introduction to expanders (11am, Fri 23 Jul 2010; notes now available)

      This is a background lecture preparing the way for the final lecture, where we
      connect results on growth in simple groups to the explicit construction of
      families of expanders. In this lecture we will define what we mean by a
      family of expanders, stating (and sometimes even proving!!) background
      results that will be important later.

      We will also try to explain why expanders are of such interest to so many
      different groups of people.

    6. Using growth results to explicitly construct expanders (1pm, Tue 27 Jul 2010; notes now available)

      We outline the method of Bourgain and Gamburd. They were the
      first to use
      results concerning growth in simple groups to explicitly construct
      expander graphs. Let S be a set in SL_2(Z) and define S_p to be the
      natural projection of S modulo p. Now let G_p be the Cayley graph of
      SL_2(p) with respect to the set S_p. Bourgain and Gamburd give precise
      results as to when the set of graphs {G_p : p a prime} forms a family of
      expanders. They make crucial use of the result of Helfgott (encountered in Seminar 2) which states that “all generating sets in SL_2(p)

Finite classical groups

This is the webpage for my course with the London Taught Course Centre. Practical matters:

  • Lectures: There are 5 lectures, given every Monday from 30th September until 28th October 2013, 3:30pm – 5:30pm at De Morgan house in London.
  • Lecture notes: These will be put online the morning after the lecture is given. Each lecture contains many exercises which students are encouraged to attempt. Answers to starred exercises will be put online after one week.
  • Discussion class: I will run an experimental online discussion class for one hour starting at 2pm on Tuesday 15th October. This is optional and will cover some extra material tangential to the course.
  • Exam: This has happened. Email me if you want solutions.

Course material follows.

Letter to the Journal of Algebra

I sent the following email to the Journal of Algebra on 16 October 2009. I know I’m not the only one who has decided to refuse to give references for Elsevier journals. (This article gives an idea why.)

Dear Prof ***,

> Manuscript Reference Number: ***
> Manuscript Title: ***

Ordinarily I would be very pleased to write a reference for this paper –
it certainly sounds like it would be of interest to me.

However I have been talking to a few people here in Bristol about journal
prices, and there is a lot of discontent at Elsevier’s policies in this
area. It has been suggested to me that there should be a referee’s boycott
of Elsevier, for as long as they continue to charge astronomical prices.
I’m inclined to agree with this position, and have taken a personal
decision to refuse requests for references for Elsevier journals.

Please be assured that my position is not meant as a criticism, in any
way, of the academics who publish in, and edit, the Journal of Algebra. It
is an excellent journal and one with which, but for this issue of pricing,
I would be pleased to be associated.

Indeed, the fact that the Journal of Algebra (amongst others) has such an
excellent academic reputation makes it all the more disappointing that its
owners choose to market it in such a mercenary way.

Very best wishes,
Nick Gill