Math 8246 Group Theory Spring Semester 2007

Instructor
Peter Webb, 350 Vincent Hall, 625 3491, webb@math.umn.edu, http://www.math.umn.edu/~webb

Office Hours
either 10:10 - 11:00 MWF or some later time (we will discuss this) or by appointment.

This semester will be devoted to the cohomology of groups.

Texts
There will be no text to purchase. I list some books below which will be helpful. The first three are expert treatments of cohomology groups, written by experts on the subject. They start from the beginning but then go into more specializes topics according to the authors' tastes, and these specialized topics are different to the ones we shall pursue. There are also other books in this category which I do not mention
K.S. Brown, Cohomology of Groups, Graduate Texts in Math. 87, Springer-Verlag 1982.
K.W. Gruenberg, Cohomological Topics in Group Theory, Lecture Notes in Math. 143, Springer-Verlag 1970.
A. Adem, R.J. Milgram, Cohomology of finite groups, Springer, 2004.

The next books I mention are general books on homological algebra. I regard them both as outstanding treatments, and the book by Weibel has the advantage of being more recent with some up-to-date topics, but we will not study these.
C.A. Weibel, An introduction to homological algebra, Cambridge University Press, 1994.
P. Hilton & U. Stammbach, A course in homological algebra, Graduate Texts in Mathematics 4, Springer 1997.

Very importantly, the books we may have used for the general algebra course have sections on homological algebra and cohomology of groups and will be useful. Because many of you will have the book by Dummit and Foote I may often make reference to it.
D.S. Dummit and R.M. Foote, Abstract Algebra, Wiley.
J.J. Rotman, Advanced modern algebra, Prentice Hall, 2002.

When we come to study crystallographic groups I will describe a theoretical approach for which the following book is useful:
H. Brown et al, Crystallographic groups of four-dimensional space, Wiley 1978.

Course Content
Basic homological algebra.
Projective resolutions, Ext, extensions of modules.
Definition of group cohomology.
Interpretations of low dimensional groups: the Schur multiplier, group extensions, the first homology and cohomology groups.
Relations with subgroups: the Schur - Zassenhaus theorem.
Applications: crystallographic groups
Further topics such as ring structure, methods of computation.

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 believe 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.
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 content of the Math 8200 algebra sequence is sufficient as a prerequisite. This semester will be largely independent of the group theory studied in the Fall Semester. The only things which will be helpful are the basic properties (existence and defining property) of free groups, and perhaps the basic material about actions of groups on sets which we did.

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.

Date of this version of the schedule: 1/15/2007