Computational group theory
Announcements
Announcements will be posted here from time to time. Please check regularly. The most recent announcement(s) will always be in green.
Lecturer
Ken Brown, Malott 521, 5-3598, kbrown@cornell.edu.
The class meets Tuesdays and Thursdays, 8:40–9:55, in Malott 205.
Course mailing list
Mail sent to math7350@math.cornell.edu will reach everyone in the class (including me). I will use this for announcements, but students can also use it for questions of general interest, discussion, etc.
Course description and prerequisite
This is a course about algorithms for computing interesting things about groups. For example, how can we compute the order of a given finite group? The answer, as we will see, depends a great deal on what the word "given" means.
To some extent I will follow the book by Holt et al mentioned below, but the precise topics will depend on the interests of the students.
The prerequisite for the course is a knowledge of basic group theory, as taught in Math 4340 or Math 6310.
Projects
There are no formal requirements for the course, but I encourage students to do a project, which will culminate in a survey lecture to the class. If you are interested, please see me to discuss possible topics.
References on reserve
Derek F. Holt, Bettina Eick, & Eamonn A. O'Brien, Handbook of computational group theory, CRC Press LLC, 2005. Free electronic access will be available shortly through the Cornell library.
Charles C. Sims, Computation with finitely presented groups, Cambridge University Press, 1994.
Electronic access to these two books should "just work" on campus. If you're off campus, either connect through the library catalogue or use PassKey.
Software
There are many computer algebra systems that can do group-theoretic computations, such as GAP and Magma. You will probably find it useful to have one available for experimentation. GAP and Magma are both installed on the Math Department computer system and can also be installed on your personal computer. GAP is free but not especially easy to install. (But I can help you.) Magma requires a license; see Steve Gaarder for details.
GAP demos
From time to time I will do a demo in class using GAP. I will post those here. You can either download them and load them into a GAP session with the "Read" command, or you can view them in your browser and copy and paste them into a GAP session. If anyone prefers a system other than GAP and wants to translate these demos into that system, I'll be glad to post those also.
- Interactive Todd—Coxeter on <a,b,c ; ac = a2, ba = b2, cb = c2>
- Random permutation group
- Sylow subgroups
- The demo related to the reverse Todd–Coxeter example did several coset enumerations using the ITC package. Here are the corresponding GAP files:
- The demo related to the Knuth–Bendix example did two computations using the KBMAG package. Here are the corresponding GAP files:
- As I explain at the end of the handout on the Knuth–Bendix example, the choice of generators in the group G of that example is based on the nilpotent quotient algorithm. The NQ package for GAP produces the maximal nilpotent quotient H of G, as shown here: In order to actually see the definitions of the generators of H, however, you have to run the standalone nq program on the following file: