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 http://www.win.tue.nl/~jpanhuis/coxeter/.

## Announcements

- The Cryptology Group at CWI organizes an introductory short course in cryptology. The target audience is 3rd year (or higher) students of mathematics and theoretical computer science. The course takes place in the mathematical institute at Leiden University on December 10,13, and 14, 2007. [More information]

## Organization

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.

## Schedule

lecture | date | subject | ||

1 | September 11 | introduction to both permutation groups and Coxeter groups | ||

2 | September 18 | permutation groups | ||

3 | September 25 | Coxeter groups | ||

4 | October 2 | permutation groups | ||

5 | October 9 | permutation groups | ||

6 | October 16 | Coxeter groups | ||

7 | October 23 | permutation groups | ||

8 | October 30 | Coxeter groups | ||

9 | November 6 | permutation groups | ||

10 | November 13 | Coxeter groups | ||

11 | November 20 | permutation groups | ||

12 | November 27 | Coxeter groups | ||

13 | December 4 | permutation groups | ||

14 | December 11 | Coxeter groups | ||

15 | December 18 | Coxeter groups |

## Homework

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.

lecture | due date | problems | ||

1 | September 18 | exercises 1 -- 29^{(1)} | ||

2 | September 25 | exercises 30 -- 43 | ||

4 | October 9 | exercises 44 -- 64 | ||

5 | October 16 | exercises 44 -- 64^{(2)} | ||

7 | October 30 | exercises 65 -- 82^{(3)} | ||

9 | November 13 | exercises 65 -- 82^{(2, 3)} | ||

11 | November 27 | exercises 83 -- 103 | ||

13 | December 11 | exercises 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.

**Remarks**

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

## Description

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').

## Literature

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.

## Examination

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.

## Prerequisites

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 http://websites.math.leidenuniv.nl/algebra/.

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