The Dutch national master course permutation groups and Coxeter groups consists of two halves: `Permutation groups' taught by H.W. Lenstra and `Coxeter groups' taught by A.M. Cohen. The general purpose of the course is to acquaint the student with several aspects of group theory.

This is the web page for the half concerning permutation groups. For the other half, see



Permutation groups: Lectures: H.W. Lenstra
Problem session: Marco Streng
Notes: Joris Weimar and Joost Michielsen
Coxeter groups: Lectures: A.M. Cohen
Problem session: Jos in't panhuis
Location:Universiteit Utrecht
Buys Ballot Laborarium room 107a
Princetonplein, Utrecht
Time:Tuesdays, September 11 - December 18, 2007, 10:15 - 13:00

During the first meeting, on 11 September 2007, each of the two instructors will give a general introduction to his subject. During the rest of the semester, alternate Tuesdays will be devoted to permutation groups and to Coxeter groups; the precise schedule will be announced in due course. Typically, there will be two hours of lectures on each Tuesday, and one hour devoted to homework problems, each hour lasting 45 minutes, with intermissions in between.


1September 11introduction to both permutation groups and Coxeter groups
2September 18permutation groups
3September 25 Coxeter groups
4October 2permutation groups
5October 9permutation groups
6October 16 Coxeter groups
7October 23permutation groups
8October 30 Coxeter groups
9November 6permutation groups
10November 13Coxeter groups
11November 20permutation groups
12November 27Coxeter groups
13December 4permutation groups
14December 11Coxeter groups
15December 18Coxeter groups


Homework problems for every lecture on permutation groups will be made available here. Students who wish to receive credit for the course must hand in solutions to four of these problems, of their own choice, before the start of next week's lecture, which can be a lecture on permutation groups or on Coxeter groups. See also `Examination'. Do not forget the homework for the other half of the course.

lecturedue dateproblems
1September 18exercises 1 -- 29(1)
2September 25exercises 30 -- 43
4October 9exercises 44 -- 64
5October 16exercises 44 -- 64(2)
7October 30exercises 65 -- 82(3)
9November 13exercises 65 -- 82(2, 3)
11November 27exercises 83 -- 103
13December 11exercises 83 -- 103(2)

Each week, you may hand in solutions to at most four exercises. Each solution is awarded with at most 3 points. The grade for an exercise set is t*9/12+1, where t is the total number of points for the exercises. If the exercises are handed in late, then the grade will be at most 7.5. You are allowed to do exercises from earlier sets as long as you have not already handed them in and the solution was not treated in class. The complete series of exercises can be found here.

  1. In exercises 28 and 29, the old version of the exercises said "p2", where it should say "p^2".
  2. Again.
  3. In exercise 81, take X finite and non-empty.


The general question treated in the course by H.W. Lenstra is to give an upper bound for the size of a finite solvable permutation group in terms of the number of elements being permuted. The main theorem to be proved has, through Galois theory, potential implications in computer algebra, specifically to the algorithmic problem of deciding whether one can express the zeroes of a given polynomial by means of radicals and, if so, actually finding such an expression. Although the nature of these implications will be explained by way of motivation, the main emphasis is on the proof of the group-theoretic theorems involved. It turns out that, even though these theorems are stated purely in terms of groups acting on sets, their proofs depend on more advanced notions from algebra, including wreath products, rings, modules, exact sequences, tensor products, induced representations, simple and semisimple modules and rings, finite fields, and extraspecial groups. An important side effect of the course will be to acquaint the student with these notions, with their properties, and how to work with them. All these notions will be defined in the course and their properties, as needed, developed, with the exception of the basic material mentioned below (see `Prerequisites').


Much of the material on permutation groups can be found in "Matrix groups" by D.A. Suprunenko (Amer. Math. Soc., 1976), but this book will not be followed in detail. Joris Weimar and Joost Michielsen, two of the students taking the course, will, soon after each lecture on permutation groups, make TeXed notes available here. Errors in the notes are to be mailed to them at jweimar (at) math (dot) leidenuniv (dot) nl.


Every week approximately ten homework problems will be handed out and published on one of the websites. Students who wish to get credit (8EC) for the course should, every week, hand in solutions to four of these problems, of their own choice. Cooperation between students is allowed (and indeed encouraged), but each student should write down the solution in his/her own words; solutions that are verbally identical are not acceptable. There will not be a final examination. The students' grades are determined by their performance on the homework.


It is supposed that the students had some previous exposure to algebra and linear algebra, including the basic properties of groups, rings, and fields. Much more than is needed from algebra can be found in the book "Algebra" by S. Lang (Springer-Verlag, 2002) as well as in the Dutch course notes Algebra 1, 2, 3 by P. Stevenhagen (Universiteit Leiden), see

Last change: .