An Introduction to Computational Group Theory

C an one rotate only one corner piece in Rubik’s cube? What are the energy levels of the buckyball molecule? Are the graphs on Figure 1 isomorphic? What is the Galois group of the polynomial x8 + 2x7 + 28x6 + 1728x + 3456? What are the possible symmetry groups of crystals? These are all questions which, directly or in a not so obvious way, lead to problems in computational group theory. Algebraic structures are well suited for machine computations. One reason for that is that we can describe large objects very concisely by a set of generators: for example, 50 bits are enough to define GL5(2), a group of order 9999360, by two 0-1 matrices of size 5× 5. Even more importantly, often we can find a generating set which reflects the structure of the group so that structural and quantitative properties can be read off easily. Computational group theory (CGT) is one of the oldest and most developed branches of computational algebra. Although most general-purpose symbolic algebra programs can handle groups to a certain extent, there are two systems which are particularly well suited for computations with groups: GAP and Magma. Also, numerous stand-alone programs and smaller systems are available. GAP can be obtained by anonymous ftp from servers on three continents; the addresses can be found on the World Wide Web page http://www-groups.dcs.st-and.ac.uk/. For the availability of Magma, please see the World Wide Web page http://www.maths. usyd.edu.au:8000/comp/magma/. The important subareas of CGT correspond to the most frequently used representations of groups: permutation groups, matrix groups, and groups defined by generators and relators, as well as to perhaps the most powerful tool for the investigation of groups, representation theory. Also, there are specialized and more efficient algorithms for special classes such as nilpotent or solvable groups. In this survey in each subarea we attempt to indicate the basic ideas and the size of jobs which can be handled by the current systems on a reasonable machine. Of course, we cannot be comprehensive here. Also, because of space restrictions, our reference list consists only of surveys, conference volumes, books, and journal special issues. Individual results are referenced in the text only if they appear in these volumes; most of the others can be traced back from these sources. An extended version of this article, with complete references, can be obtained from http://www.math.ohiostate.edu/ ̃akos/ or http://www.math. rwth-aachen.de/ ̃Akos.Seress/. E. O’Brien’s database of papers on group theory, including Ákos Seress is associate professor of mathematics at The Ohio State University, Columbus, Ohio. His e-mail address is [email protected].