Computational Group Theory at Colorado State University

Computational Group Theory (CGT) is the study of algorithms, theoretical as well as for concrete calculations, for computing with groups. Such algorithms find use both in group theory itself and in areas (such as combinatorics, topology or physics) that use groups to describe symmetries.

Colorado State University at Fort Collins — the American center for the development of the system GAP —has an active research group in CGT.

You can also find course notes for a recent graduate course here.


Anton Betten
Alexander Hulpke


Kenneth Monks
Ellen Ziliak
Andreea Erciulescu

Former Students

Soley Jonsdottir



Research in Fort Collins covers many aspects of CGT. The following list gives some areas on which work has been done recently or is continuing:

Some relevant publications can be found on this page.

Recent and current student and thesis projects include:


Computational Group Theory uses methods from Algebra and Combinatorics. If you are interested in doing research in this area, be it as undergraduate research, as masters thesis or as a doctoral thesis project, the following list gives some information on courses and prerequisites. Feel free to contact me if you have further questions.

Undergraduates: You should take (in particular) M366 and M369.

Graduates: Typically incoming graduate students will start with the first-year courses on Algebra (M566/567) and Combinatorics (501/502) and continue with the second year courses (offered only every second year) on representation theory (M666). The courses on advanced combinatorics (M601/602) and commutative algebra (M667) are strongly recommended.
If you did not have linear algebra as an undergraduate (or just a short course) you should also consider M560.

Relevant topics courses, offered if there is sufficient student interest (M676), include Computational Group theory, Computer Algebra and Algebraic Number Theory.

Prospective Students: If you are not yet a student at Colorado State University you can find information about our programs on the departmental web pages.

Many projects involve explicit computation. For these, basic knowledge of some programming language is helpful, but is not a prerequisite.

Potential Projects: I maintain a list of GAP-related thesis and project topics, but this is not considered exclusive.


Information Technology Aspects

While the development of algorithms for groups is the majority of mathematical work (and requires knowledge of graduate level mathematics), the fact that algorithms end up in a distributed system provides a series of challenges that are much closer to computer science and engineering:

Alexander Hulpke (, January 20, 2009