Math 8245 Group Theory Fall Semester 2003

Instructor
Peter Webb, 350 Vincent Hall, 625 3491, webb@math.umn.edu, http://www.math.umn.edu/~webb
Office Hours
11:15-12:05 MWF or by appointment.

Text
There will be no text to purchase. Lecture notes on representation theory and some GAP tutorials will be available from http://www.math.umn.edu/~webb
Information about GAP is obtained from its web site http://www.gap-system.org/
From this site you can download GAP free of charge to your own computer.
From this page click on 'GAP support', then 'manual', then 'tutorial' to get a tutorial (but we will not use this tutorial in class).
From the same page as before click on 'About GAP', then 'examples', then 'Rubik's cube' to get a useful example (which we will also probably not use in class).

Other useful texts
DJS Robinson, A course in the theory of groups, 2nd edition, Springer-Verlag, ISBN 0387944613.
J.L. Alperin, Local representation theory, Cambridge University Press 1993, ISBN 052144926X.
J.-P. Serre, Linear representations of finite groups, Springer.
A.M. Cohen et al, Some tapas of computer algebra, Springer 1999, ISBN 3540634800 (chapter 8, projects 5 and 6).
D.L. Johnson, Presentations of groups, Cambridge University Press 1990, ISBN 0521378249, chapters 8 and 9.
C.C. Sims, Computation with finitely presented groups, Cambridge University Press 1994, ISBN 0521432138
J. Neubueser, An elementary introduction to coset table methods in computational group theory, pp. 1-45 in Groups - St. Andrews 1981 (C,M, Campbell and E.F. Robertson, eds), Cambridge UP.


Course Content
For the first 10 weeks or so of this semester there will really be two parallel courses, one addressing the representation theory of finite groups, the other dealing with computational group theory using the computer system GAP. Representation theory will be taught on 2 days each week (Monday and Wednesday), and on the third class day each week (Friday) we will meet in the computer lab Vincent Hall 5. For this you will need an account on the Math Dept computers. No prior programming experience is necessary. We will start by learning the basics of the language which GAP uses, and go on to learn how to do computations with groups given as permutation groups, by presentations, and as matrix groups. As part of this we will learn how the algorithms we use work, and thereby gain some insight into their limitations.

The representation theory to be taught will not be like other classes on representation theory on offer. The approach will be to teach the theory in a characteristic-free manner where possible and both the characteristic zero theory (character tables, orthogonality, Burnside's theorem...) and also the positive characteristic theory (indecomposable modules, injectives and projectives, blocks, vertices and sources...) will be taught. I am particularly keen to teach the positive characteristic theory because in my experience many people feel less confident about it, and it has applications in many parts of pure mathematics, including topology, combinatorics, number theory as well as to other parts of group representation theory.

Course Assessment
I will assign a set of homework problems roughly every 2 weeks, giving a total of six homework assignments altogether. If you make a genuine attempt at 50% or more of the questions you will get an A for the course. You do not have to obtain correct solutions to these questions, only make genuine attempts (in my opinion). I am well aware that the existence of homework in advanced courses is considered by students quite negatively. I believe myself that it is extremely difficult to obtain a sound and permanently-lasting command of the material presented without doing some work which actively involves the student. If people wish to propose other forms of active involvement, I will be extremely willing to discuss these. As it is, it should be possible for everyone who wishes to obtain an A on this course.

Expectations of written work
Most of the time in the conventional homework problems, to satisfy my criterion of making a genuine attempt you will need to write down explanations for the calculations and arguments you make. Where explanations need to be given, these should be written out in sentences i.e. with verbs, capital letters at the beginning, periods at the end, etc. and not in an abbreviated form.
Some of the homework will be computer exercises in GAP. An essential part of what you hand in for these exercises will be a transcript of a GAP session, but it will help if you insert explanatory comments. Your work will exist as a computer file, and it would be possible to send it to me by email, but I do prefer to see a hard copy of what you have done. I have two reasons for this. One is that I find the hard copy easier to read, and the other is that with the state of email these days I find it hard to extract the messages I am really interested in from the spam.
I encourage you to form study groups. However everything to be handed in must be written up in your own words. If two students hand in identical assignments, they will both receive no credit.

Prerequisites
The only prerequisite is the content of the Math 8200 algebra sequence. As far as group theory is concerned, the topics I expect you to know include

Lagrange’s theorem, the isomorphism theorems, direct products of groups, properties of permutations (they may be written as products of disjoint cycles, conjugacy classes in the symmetric groups, the sign), structure of finitely-generated abelian groups, Jordan-Hölder theorem). Sylow’s theorems, simplicity of A₅, beginning facts about solvable groups, p-groups.

Incompletes
These will only be given in exceptional circumstances. A student must have satisfactorily completed all but a small portion of the work in the course, have a compelling reason for the incomplete, and must make prior arrangements with me for how the incomplete will be removed, well before the end of the quarter.

Proof-reading the representation theory notes
The notes which I hand out to you may one day be a book. I will be very appreciative of any comments you have about what I write. These may be lists of typographical errors, mathematical errors, or comments that perhaps more explanation would be in order in such and such a place, more background should be given as motivation etc. etc. In advance I give you my thanks.

Date of this version of the schedule: 9/1/2003