seress.qxp 5/21/97 4:13 PM Page 671 An Introduction to Computational Group Theory Ákos Seress an one rotate only one corner piece in groups to a certain extent, there are two systems Rubik’s cube? What are the energy which are particularly well suited for computa- levels of the buckyball molecule? Are tions with groups: GAP and Magma. Also, nu- the graphs on Figure 1 isomorphic? merous stand-alone programs and smaller sys- What is the Galois group of the poly- tems are available. Cnomial x8 +2x7 +28x6 + 1728x + 3456? What GAP can be obtained by anonymous ftp from are the possible symmetry groups of crystals? servers on three continents; the addresses can These are all questions which, directly or in a not be found on the World Wide Web page so obvious way, lead to problems in computa- http://www-groups.dcs.st-and.ac.uk/. tional group theory. For the availability of Magma, please see the Algebraic structures are well suited for ma- World Wide Web page http://www.maths. chine computations. One reason for that is that usyd.edu.au:8000/comp/magma/. we can describe large objects very concisely by The important subareas of CGT correspond a set of generators: for example, 50 bits are to the most frequently used representations of enough to define GL5(2), a group of order groups: permutation groups, matrix groups, and 9999360, by two 0-1 matrices of size 5 5. groups defined by generators and relators, as Even more importantly, often we can find a gen-× well as to perhaps the most powerful tool for the erating set which reflects the structure of the investigation of groups, representation theory. group so that structural and quantitative prop- Also, there are specialized and more efficient al- erties can be read off easily. gorithms for special classes such as nilpotent or Computational group theory (CGT) is one of solvable groups. In this survey in each subarea the oldest and most developed branches of com- we attempt to indicate the basic ideas and the putational algebra. Although most general-pur- size of jobs which can be handled by the current pose symbolic algebra programs can handle 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 Ákos Seress is associate professor of mathematics at The Ohio State University, Columbus, Ohio. His e-mail ad- journal special issues. Individual results are ref- dress is [email protected]. erenced in the text only if they appear in these volumes; most of the others can be traced back Partially supported by NSF Grant CCR-9503430 and by the Alexander von Humboldt Foundation. from these sources. An extended version of this article, with complete references, can be ob- Acknowledgement: the author is indebted to G. Havas, tained from http://www.math.ohio- D. Holt, W. Kantor, K. Lux, J. Neubüser, and E. O’Brien for their helpful comments. The part of the section state.edu/˜akos/ or http://www.math. “Polyclyclic Groups” about quotient group methods was rwth-aachen.de/˜Akos.Seress/. E. O’Brien’s written by J. Neubüser. database of papers on group theory, including JUNE/JULY 1997 NOTICES OF THE AMS 671 seress.qxp 5/21/97 4:13 PM Page 672 Finitely Presented Groups Let G = E be a presentation for a h |Ri group G : E = g1,... ,gn is a finite set of { } generators, and = r1 =1,... ,rm =1 is R { } a set of defining relations. Each ri is a word, using the generators in E and their inverses. The basic questions are to de- cide whether G is finite and to determine whether a given word represents the iden- tity of G. By the celebrated result of Novikov and Boone, these questions are undecidable: Figure 1. they cannot be answered by a recursive al- gorithm. Nevertheless, because of the prac- a lot of references to CGT, is available via tical importance of the problem, a lot of effort http://www.math.auckland.ac.nz/ is devoted to the development of methods for ˜obrien/. investigating finitely presented groups. We start with some historical remarks. Algo- One basic method is the Todd-Coxeter coset rithmic questions permeated group theory from enumeration procedure. Given G = E and h |Ri its inception in the last century. As examples, H = h1,... ,hk , where H G and each hj is a consider that Galois’s work was inspired by the wordh in the generatorsi of G≤and their inverses, solvability of equations. Thirty years later, Math- our goal is to compute the permutation repre- ieu announced the existence of the 5-transitive sentation of G on the right cosets of H. group M24, but he needed twelve years to find We set up a coset table: this is a matrix M, the “clarity and elegance necessary to present it.” where the rows are labelled by positive integers, Had he access to a computer, this period could representing cosets of H, and the columns are probably have been shortened significantly. Jor- labelled by the elements of 1 1 E := g1,... ,gn,g− ,... ,g . The entries (if dan, Hölder, Cole, and others could also have { 1 n− } used the machine in their quest to classify groups defined) are positive integers, M(k, g)=l, if we of small order. know that kg = l for the cosets k, l and for The “official” starting date of CGT may be g E . Originally, we have a 1 E table with no∈ entries, where 1 denotes the×| coset| H 1. As pinned down in 1911, when Dehn proposed the · solution of the word problem, Find an algorithm new cosets are defined, we add rows to the coset table. to decide whether, in a group defined by a finite Of course, we have to detect when two words, set of abstract generators and relators, a word defining different rows of the table, actually be- in the generators represents the identity. Dehn’s long to the same coset of H. To this end, for each question was motivated by topological consid- relation ri = gi gi gi , we also maintain a re- erations; even today it is hard to draw a sharp 1 2 ··· t lation table. This is a matrix Mi, with rows la- border between combinatorial group theory and belled by the cosets 1, 2,... , as defined in M, topology. The flourishing of CGT started in the and columns labelled by the elements of the se- sixties, when, for example, the basic methods for quence (gi ,gi ,... ,gi ). The entry Mi(k, gi ), if permutation group manipulation and the com- 1 2 t j defined, is the number of the coset kgi gi . putation of character tables were established, 1 ··· j Initially, we have Mi(k, git )=k for each row num- and term rewriting procedures were introduced. ber k, since ri =1 in G. Whenever a new coset Not much later, the first large applications, such is defined, we fill all entries of the relation ta- as Sims’s existence proof for Lyons’s sporadic bles that we can. simple group, arose, and the development of Finally, for each generator hj = gj1 gjt of the first integrated system, the Aachen-Sydney H, we maintain a subgroup table. This··· is a ma- Group System, started. Since then the area has trix Sj with only one row, corresponding to the been growing rapidly, both in terms of the design, coset H 1, and columns labelled by the factors · implementation, and application of algorithms, of hj. The rule for filling entries is the same as as well as in the number of mathematicians in- for the Mi ; originally, Sj (1,gjt )=1, since volved in this development. Nowadays, some of Hhj = H. the major lines of development are the integra- When the last entry is filled in a row of a re- tion of consequences of the classification of fi- lation table or a subgroup table, we also get an nite simple groups and methods suggested by extra piece of information, kg = l, for some complexity theoretical considerations into prac- cosets k, l and g E . This is called a deduction. ∈ tical algorithms and the systematic use of ran- If the entry M(k, g) is not yet defined, then we 1 domization. A more detailed history is in [11]. fill the entries M(k, g), M(l,g− ), and all possi- 672 NOTICES OF THE AMS VOLUME 44, NUMBER 6 seress.qxp 5/21/97 4:13 PM Page 673 ble entries in the relation and subgroup tables; this way, we may get further deductions. If M(k, g) is already defined but l := M(k, g) = l, then we 0 6 realize that l,l0 denote the same coset of H. This Figure 2. is called a coincidence. We replace all occurrences of l,l0 by the smaller of An alternative method to coset enumeration these two numbers and fill the entries of the ta- is the Knuth-Bendix term-rewriting procedure bles that we can. This may lead to further de- [13]. We collect a list of pairs of words (u, v) such ductions and coincidences. The process stops that u, v represent the same element of G. These when all entries of the coset table, the relation pairs are called rewriting rules, since we can re- tables, and subgroup tables are filled. place a word w1uw2 by w1vw2. The goal is to collect a confluent system of rules: no matter in We illustrate these ideas by enumerating which order the rules are applied, every word in G = g ,g g2 =1,g2 =1, (g g )3 =1 = S on 1 2 1 2 1 2 ∼ 3 the generators is converted into a unique nor- the cosetsh |of the subgroup H = g gi g g of 1 2 1 2 mal form.