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. Although the usual problems of un- order 3. Since both generators areh involutions,i decidability arise, Knuth-Bendix methods can we have E = E. Also, we maintain only one rela- 3 sometimes solve the word problem for infinite tion table, corresponding to (g1g2) =1; the other two relators tell us that at the definition groups, which can almost never be done by coset of new cosets, we should multiply previous enumeration techniques. An interesting recent development is the de- cosets by g1,g2 alternatingly. Figure 2 shows the coset table CT, relation table RT, and subgroup finition and the algorithmic handling of auto- table ST after the definition of the cosets matic groups. These are groups with solvable word problem and include important group 1:=H,2:=1g1, 3:=1g2, 4:=2g2 . At that mo- ment, the last entry (in the second column) of classes occurring in topology, such as the fun- damental groups of compact hyperbolic and Eu- ST is filled and we get the deduction 4g1 =3, clidean manifolds and of hyperbolic manifolds which also implies 3g1 =4. Then all entries in CT are known, and we can complete RT; this of finite volume, word hyperbolic groups, and leads to the coincidences 1=4and 2=3. groups satisfying various small cancellation properties. Taking minimal care in defining new cosets If a presentation for a subgroup H G is ≤ (namely, if a coset k is defined, then sooner or needed, Schreier’s subgroup lemma may be used later we define kg for all g E ), it is guaranteed to obtain generators for H. that the algorithm terminates∈ if G : H < . However, there is no recursive function| of |G : H∞ Lemma 1.1. Let T be a right transversal for H | | in G = S , and, for g G, let g denote the el- and the input length which would bound the h i ∈ ement of T such that g Hg. Then number of cosets defined during the procedure. ∈ It is easy to give presentations of the trivial 1 tsts− : t T,s S group such that no commonly used variant of { ∈ ∈ } the Todd-Coxeter algorithm can handle them. generates H. This, and different coset enumeration strate- Using the fact that words representing sub- gies, are discussed, for example, in [13, Ch. 5]. group elements can be rewritten as products of A very accessible, elementary description of the Schreier generators, one may obtain a presen- methods is in [10]. Success mostly depends on tation for H [13, Ch. 6]. This presentation is the number of entries defined in the coset table usually highly redundant, and it can be shortened rather than G : H . There are instances of suc- by applying the so-called Tietze transformations cessful enumeration| | with G : H > 106. | | [13, Ch. 1]. It is often worthwhile to do Tietze If we do not have a candidate for a small transformations interactively, guiding the com- index subgroup H in G, we may try programs puter to the type of presentation we try to which find some or all subgroups of G with achieve. index at most a given bound n. These programs consider all coset tables with at most n rows and Polycyclic Groups use a backtrack search to eliminate those which In the rest of this survey, we shall almost ex- are not consistent with the given relators. De- clusively deal with groups for which the unde- pending on the complexity of the presentation, cidability of the word problem vanishes. This will in some cases we can expect success for values clearly be the case with finite groups given, for of n up to about 100. example, as permutation or matrix groups, but

JUNE/JULY 1997 NOTICES OF THE AMS 673 seress.qxp 5/21/97 4:13 PM Page 674

it is also the case for groups given by polycyclic the newly obtained vectors have to be formed, presentations. corresponding to the construction of Schreier A group is called polycyclic if it has a finite generators in Lemma 1.1.) Then each h H can ∈ subnormal series of the form be written uniquely in the form

di1 di2 dik (1) G = G1 B G2 B B Gn B Gn+1 =1, (3) h = h h h , ··· 1 2 ··· k

with cyclic factors Gi/Gi+1. If, for 1 i n, we with exponents 0 di < Gi : Gi . The pro- ≤ ≤ ≤ j | j j+1 | pick ai Gi Gi+1 such that Gi = Gi+1,ai , cedure can be used for membership testing in ∈ \ h i then each g G can be written uniquely in the H as well, by attempting to factorize any given e1 ∈e2 en form g = a a an , where ei Z for infinite g G as in (3). When membership testing is 1 2 ··· ∈ ∈ cyclic factors Gi/Gi+1, and 0 ei < Gi : Gi+1 available, we can compute normal closures of ≤ | | for finite factor groups. This description of g is subgroups and derived series and lower central called a collected word for g. For finite poly- series, and handle homomorphisms. This cyclic groups, we also have method can be extended to infinite polycyclic groups, and we shall see an analogous procedure Gi :Gi+1 εi;i+1 εi;i+2 εi;n ai| | = ai+1 ai+2 an , for permutation groups in the next section. ··· The other basic method is to consider larger for 1 i n, (2) ≤ ≤ and larger factor groups of G. We construct a εi,j;i+1 εi,j;i+2 εi,j;n aj ai = aia a an , subnormal chain (1), which is a refinement of a i+1 i+2 ··· for 1 i

674 NOTICES OF THE AMS VOLUME 44, NUMBER 6 seress.qxp 5/21/97 4:13 PM Page 675

first step, an epimorphism ϕ : G G/G is con- last five years, a remarkable convergence of the → 0 structed. A polycyclic presentation for G/G0 can approaches occurred: for example, in GAP, most be obtained by adding to the relators of G that of the permutation group library uses imple- all pairs of generators commute and computing mentations of algorithms with the fastest known the cyclic decomposition of the resulting abelian asymptotic running times. group. Suppose that an epimorphism ϕ is known The basic ideas of permutation group ma- onto a nilpotent or polycyclic group H. Then an nipulation are due to Sims. A base for attempt is made to lift ϕ to an epimorphism of G Sym( ) is a sequence B =(β1,... ,βm) of ≤ G onto an extension group K of H by a normal points from such that the pointwise stabilizer subgroup N that is central in the case of pQ and GB =1. TheΩ sequence B defines a subgroup NQ or just abelian in the other cases. chain Ω The methods for finding such N and K and for lifting ϕ differ in the various quotient meth- (4) G = G[1] G[2] G[m] G[m+1] =1 ods. In the case of pQ, they amount to solving ≥ ≥···≥ ≥ [i] linear equations over GF(p) that on one hand rep- where G = G(β1,... ,βi 1). A strong generating set resent confluence conditions for the pcp pre- (SGS) relative to B is a −generating set S for G with sentation and on the other hand stem from the the property that relations for G. In the case of NQ, congruences occur instead of equations, while in the case of (5) S G[i] = G[i] for 1 i m +1. h ∩ i ≤ ≤ SQ’s, modular representations of H are con- G[i] sidered. Finally, in the case of PCQ’s, Gröbner Given an SGS, the orbits βi and transversals [i] [i+1] basis techniques come into play. It should be no Ri for G mod G can be easily computed. As wonder that the time requirement for NQ, and an analogue of (3) in the section “Polycyclic especially for SQ and PCQ, is much higher than Groups”, every g G can be written uniquely in ∈ for pQ, and so applications of these are far the form more restricted. (6) g = rmrm 1 r1 Permutation Groups − ··· with ri Ri. Factorizing elements in this form The situation is quite satisfactory in the case of ∈ permutation group algorithms as well: most is called sifting and may be considered as a per- structural properties of permutation groups of mutation group version of Gaussian elimina- reasonable degree and order can be readily com- tion. It can be used for membership testing; puted. Working with permutation representa- computing normal closures of subgroups, de- tions is usually the method of choice when study- rived series, and lower central series; handling ing simple groups and their subgroup structure, homomorphisms; and finding the pointwise sta- provided that the simple group has a permuta- bilizer of any subset of . tion representation of sufficiently small degree. We sketch the basic idea of an SGS construc- “Reasonable” and “sufficiently small” degree, of tion. Given G = T , weΩ maintain a sequence h i course, depend on the available software and B =(β1,... ,βk) of already-known elements of a hardware; currently, it is in the low hundred base and approximations Si for a generating set

thousands, while the logarithm of the group of the stabilizer G(β1,... ,βi 1). We say that the data − order may be in the low hundreds. structure is up-to-date below level j if This is the area of CGT where the complexity analysis of algorithms is the most developed. The (7) Si β = Si+1 h i i h i reason for that is the connection with the cele- holds for all j*Isomorphism Problem. The decisive ≤ result in establishing the connection is Luks’s is an SGS for G if and only if it generates G Sand polynomial time algorithm for the isomorphism the data structure is up-to-date below level 0. testing of graphs with bounded valence, where In the case when the data structure is up-to- the isomorphism problem is reduced to finding date below level j, we compute a transversal Rj setwise stabilizers of subsets in the permutation for Sj mod Sj β . Then we test whether (7) h i h i j domain of groups with composition factors of holds for i = j. By Lemma 1.1, this can be done bounded size. This result not only established by sifting the Schreier generators obtained from a link between complexity theory and CGT but Rj and Sj in the group Sj+1 . If all Schreier gen- h i provided new methodology for permutation erators are in Sj+1 , then we say that the data h i group algorithms. structure is up-to-date below level j 1; other- − Because of the approach from two different wise we add a nontrivial Schreier generator h to points of view, complexity theoretic and practi- Sj+1 and say that the data structure is up-to-date cal, algorithms for numerous tasks were devel- below level j +1. In the case j = k, we also choose oped independently in the two contexts. In the a new base point βk+1 from the support of h.*

*JUNE/JULY 1997 NOTICES OF THE AMS 675 seress.qxp 5/21/97 4:13 PM Page 676*

*We start the algorithm by choosing β1 puting the centralizer of elements, the inter- ∈ which is moved by at least one generator in T section of two subgroups, or setwise stabilizers and setting S1 := T. At that moment, the dataΩ of subsets of the permutation domain. These structure is up-to-date below level 1; the algo- tasks are polynomially equivalent, and graph rithm terminates when the data structure be- isomorphism can be reduced to them (see Luks comes up-to-date below level 0. in [6]). Therefore, it is suspected that no poly- A second generation of algorithms uses di- nomial time algorithms exist. vide-and-conquer techniques by utilizing the im- Practical algorithms for these problems use primitivity block structure of the input group, backtrack methods. Let B be a fixed base of G. thereby reducing the problems to primitive The images of base points uniquely determine groups. We mention Luks’s and P. Neumann’s re- the elements of G, while the image of an initial sults on computing a composition series, the segment of B defines a coset of some G[i] (cf. Sylow subgroup constructions by Kantor, and the (4)). The images of all initial segments define a asymptotically fastest deterministic SGS con- partial order by inclusion, which is called the struction by Babai, Luks, and Seress. The latter search tree for G. Traditional backtrack meth- runs almost a factor n2 faster than the asymp- ods systematically examine the elements of the totically fastest versions of Sims’s original search tree. Especially valuable is when a node method. The required group-theoretic arsenal for close to the root can be eliminated, because all these algorithms includes consequences of the elements of G less than this node (i.e., elements classification of finite simple groups and, in the of the appropriate coset) can be excluded at case of the Sylow subgroup constructions, de- once. A new generation of backtrack methods tailed knowledge of the classical simple groups. was developed by Leon in [3], based on ideas (We mention, however, that a composition series from B. McKay’s graph isomorphism testing pro- can be computed without using the simple group gram nauty. Nowadays centralizers can be com- classification by a result of Beals.) puted in groups of degree in the tens of thou- The running time of most algorithms for sands, provided that they have a small base. G = S Sn does not depend only on the input The computation of normalizers of subgroups h i≤ length S n but also on the order of G. More- seems to be harder: in theory, polynomial time | | over, in groups of current interest, it frequently equivalence with centralizers is not known, and happens that the degree of G is in the tens of practical computation is much more compli- thousands or even higher, so even a (n2) al- cated. gorithm may not be practical. On the other hand, log G is often small. Therefore, a recentΘ trend Representation Theory | | is to search for algorithms with running time of As John Conway writes in the introduction of the the form O(n S logk G ). These are called Atlas [5], “the ordinary character table is be- | | | | nearly linear algorithms, since their running yond doubt the most compendious way of con- time is O(n logc n) if log G is bounded from veying information about a group to the skilled | | above by a polylogarithmic function of n. This reader.” However, computing a character table happens, for example, for all permutation rep- from scratch is not an easy task. Therefore, GAP resentations of finite simple groups except the contains tables for the most frequently used alternating ones. groups, including all tables in the Atlas and ta- There is a large library of nearly linear algo- bles for most of the maximal subgroups of spo- rithms by Babai, Beals, Cooperman, Finkelstein; radic groups. Luks and Seress in [7]; Morje in [7]; and Schön- A basic idea for computing the complex ir- ert. Roughly, everything that can be done in reducible characters of a group G is due to polynomial time can also be done in nearly lin- Burnside. In the group algebra CG of G, the ear time in groups which do not have composi- sum SC of the elements of a conjugacy class C tion factors of exceptional Lie type. The price we is in the center of CG; in fact, taking these sums pay for the speedup is that most algorithms are for all conjugacy classes, we get a basis for random Monte Carlo: they may return an incor- Z(CG). For a fixed conjugacy class C, the prod-*

* rect answer, with probability of error bounded ucts SC SCi decompose as a linear combination*

* by the user. Although there is a recent theoret- of the SCj’s, and the coefficients, taken for all*

* ical result by Kantor and Seress that upgrades possible Ci, define a matrix MC. Since the SCj’s*

* all nearly linear time algorithms to Las Vegas (i.e., are in Z(CG), the matrices MCj commute and are to guaranteed correct output) for groups with no simultaneously diagonalizable. The irreducible exceptional Lie type composition factors, the characters can be easily computed from the en- practicability of this method is not demonstrated tries in the diagonal forms. yet. To perform the computation sketched above, At present there are no polynomial time al- we must know the conjugacy classes and the gorithms for some important tasks such as com- class-multiplication coefficients, which boils*

*676 NOTICES OF THE AMS VOLUME 44, NUMBER 6 seress.qxp 5/21/97 4:13 PM Page 677*

* down to an algorithm to determine which con- The other large area of representation theory jugacy class a given group element belongs to. is the theory of modular representations. Brauer Dixon performs the diagonalization over prime characters are even harder to compute than or- fields Zp for (large) primes p, and the result is dinary ones, so GAP also contains a library of lifted back to C. Schneider in [2] introduces Brauer character tables, including all tables in the*

* strategies to select a small collection of the MCj’s modular extension of the Atlas [8]. from which the new basis for the diagonal form In the case of representations over the com- can be computed. The Dixon-Schneider method plex numbers, most computations deal only with can be used for groups of moderate order (about the character tables. On the other hand, over fi- G < 109) and with not too many conjugacy nite fields, the emphasis is also on the compu- | | classes (in the low hundreds). tation with the representations themselves. One For larger groups, we may try to compute the of the reasons is that modular representations character table interactively (see Neubüser, occur naturally in different areas of CGT: for ex- Pahlings, Plesken in [1], and [9]). We may know ample, when considering groups acting on the some (not necessarily irreducible) characters elementary abelian factors (section “Polycyclic from a matrix or a permutation representation Groups”) or when studying matrix groups (sec- of G, and there are machine commands to in- tion “Matrix Groups”). duce characters of subgroups, restrict characters Dealing with a modular representation G GLd(F), a fundamental problem is to find in- of overgroups, and extend characters of factor ≤ groups. We may also take products of known variant submodules or to prove that G acts ir- characters (corresponding to tensor products reducibly. This can be done, for example, using of representations), and, if the power maps for a generalization by Holt and Rees of R. Parker’s conjugacy classes are known, we can compute idea, the so-called Meat-Axe. Also, in studying the the generalized characters defined by structure of large dimensional modules, some- times the condensation method (see Ryba in [2], χ(n)(g):=χ(gn). and [9]) can be applied. This means that given Once a set of characters is collected, the hunt G GLd(F), we find a subalgebra A FG and an for the irreducible ones may begin. Taking scalar ≤ ≤ A-module N of possibly much smaller dimen- products, we may subtract multiples of the sion than M such that the submodule lattices of known irreducible characters to decrease the M and N are isomorphic. For example, Cooper- norms of the characters in our collection. Ear- man, Hiss, Lux, and Muller recently condensed lier methods tried to split characters as the sum a 976841775-dimensional module of the spo- of characters with smaller norms. Nowadays the radic simple Thompson group to 1403 dimen- most effective method seems to be the applica- sions. tion of Lovász’s basis reduction algorithm to the set of known characters to obtain a basis of Matrix Groups the lattice of the generalized characters which L Matrix groups are important and very compact consists of elements of small norm. Then, fol- representations of groups, but pose serious com- lowing an idea of W. Plesken, we may consider putational problems. For groups over the inte- embeddings of the basis vectors of into the gers, the membership problem is undecidable al- m L lattice Z as vectors with the indicated norm and ready in four dimensions. (However, on the m find the standard basis of Z , which corre- positive side, the finiteness of matrix groups sponds to the irreducible characters, as linear over Z can be determined by a practical poly- combinations of the basis vectors of . Quite re- L nomial time algorithm of Babai, Beals, and Rock- markably, despite the numerous possibilities more.) There are difficulties with matrix groups for the embedding, this method usually quickly defined over finite fields as well. Rewriting ma- produces some irreducible characters. trix groups as permutation groups on the un- The method of generating characters by hunt- derlying vector space, which was the early ap- ing for irreducibles can be iterated: when a larger proach, results in an exponential blowup of the collection of irreducible characters is known, it input size. Moreover, two basic ingredients, may be worthwhile to induce characters of which were crucial for the efficient handling of smaller subgroups or compute higher tensor solvable and permutation groups, are missing: powers. Initially, we may restrict the computa- there may be no subgroup chain with small in- tions to rational characters because of the cost dices, analogous to (1) or (4), and it is not clear of handling the irrational numbers. When irra- how to generalize Gaussian elimination. Large tional characters are finally computed, we can fields also cause problems, since the discrete log- use the fact that if n is the order of some g G, arithm problem arises already for 1 1 matri- ∈ × then each character value on g can be written ces. Nevertheless, despite all these difficulties, as an integer linear combination of the powers there is a concentrated attack from both the of a primitive nth root of unity. practical and theoretical sides to determine the*

*JUNE/JULY 1997 NOTICES OF THE AMS 677 seress.qxp 5/21/97 4:13 PM Page 678*

* structure of matrix groups over finite fields. Concerning the other cases of the Aschbacher Currently, this is the most active area of CGT. classification, the Meat-Axe (cf. section “Repre- Most effort concentrates on the matrix recog- sentation Theory”) can be used to test irre- nition project: given a matrix group G GL (q) ducibility and absolute irreducibility of G. There ≤ d by a list of generators, find at least one category are also programs by Holt, Leedham-Green, of Aschbacher’s classification to which G be- O’Brien, and Rees computing imprimitivity and longs. Aschbacher’s theorem classifies sub- tensor decompositions. The exploration of groups beyond classifying them in one of the As- groups of GLd(q) into nine categories: roughly, modulo scalars G is an almost simple group or chbacher classes has also started. The imple- mentations of the algorithms indicated in the isomorphic to a subgroup of GLd(q0) for some q q, or there is a normal subgroup of G natu- previous three paragraphs can handle groups of 0| dimension in the low hundreds over moderately rally associated with the action of G on the un- sized fields (q<216). derlying vector space. Once a category is found, While randomization can often be used to it can be used to explore the structure of the speed up permutation group algorithms, it seems group further. For example, if the group acts im- to be an indispensable tool for the study of ma- primitively and we find a decomposition trix groups. In theory, concerning polynomial V = V1 V2 Vm of the underlying vector ⊕ ⊕···⊕ time construction the situation is satisfactory: space such that the V are permuted by G, then i Babai gives a Monte Carlo algorithm which, after we can construct a homomorphism ϕ : G Sm. 5 → a preprocessing phase consisting of O(log G ) The image of ϕ is a permutation group of low | | group operations, constructs independent, degree, so it can be handled easily. In this case, nearly uniformly distributed elements at a cost the “naturally associated” normal subgroup is the of O(log G ) group operations per random ele- | | kernel of ϕ; we can obtain generators for it and ment. (Nearly uniform means that each element apply our methods recursively to the action on of G is selected with probability between the lower-dimensional vector spaces Vi. (1 ε)/ G and (1 + ε)/ G for some small ε.) In − | | | | The first subproblem solved both in theory practice, the heuristic product replacement al- and practice is the recognition of the classical gorithm of Celler, Leedham-Green, Murray, almost simple groups in their natural represen- O’Brien, and Niemeyer is used. This starts with tation. Neumann and Praeger introduced the a list (g1,g2,... ,gm) of generators; at each step, basic ideas. We say that a number is ppd(q,d) two indices i,j are selected randomly and one if it has a prime divisor p such that p qd 1 and | − of gi,gj is replaced by the product gigj. After p qi 1 for 1 i d 1. Neumann and 6| − ≤ ≤ − K replacements as preprocessing, we start to out- Praeger give precise estimates for the proportion put the newly created elements of the list as ran- of elements in groups G SL (q) whose order ≥ d dom elements of G. It is shown that if m is at is ppd(q,d) and for those whose order is least twice the size of a minimal generating set ppd(q,d 1). It turns out that among 5.5d ran- of G, then the limit distribution of the sequence − domly and independently chosen elements, both (g1,g2,... ,gm) is uniform among all generating types occur with probability >0.99. On the other sequences of length m, and computational evi- hand, the (short) list of subgroups containing dence shows that in applications for matrix both types of elements but not containing SLd(q) groups of dimension in the low hundreds, K is determined, and special tests are devised to can be chosen around 100. Although the con- eliminate them. This method was extended by structed random elements are not independent, Niemeyer and Praeger in a highly nontrivial way the ratio of elements with properties relevant for to the other classical groups. An alternative ap- the algorithms (for example, the ppd property) proach is described by Celler and Leedham- seems in experimental tests to be close to the Green in [7]. Here, random elements of G are cho- ratio of such elements in the group. sen, their order is determined, and groups of the We mention that some of the algorithms de- Aschbacher classification which cannot contain scribed in this section work in the more general elements of this order are eliminated. context of black box groups (cf. the extended ver- The classical matrix groups can also be rec- sion of this survey on the World Wide Web). ognized constructively; the output is not only a proof that G is a classical group, but every ele- Applications ment of G can be expressed effectively in terms We finish this survey by briefly mentioning some of the given generating set. Celler and Leedham- available databases, applications of CGT, and Green handle the case G SL (q) in the natural stand-alone programs. ≥ d representation; Cooperman, Finkelstein, and Lin- In the nineteenth century a central problem ton in [7] handle the case G =SLd(2) in any rep- of group theory was the classification of groups resentation; while Kantor and Seress handle all of a given order. Currently, GAP contains a library classical groups in any representation. of the 174366 groups of order at most 1000, but*

*678 NOTICES OF THE AMS VOLUME 44, NUMBER 6 seress.qxp 5/21/97 4:13 PM Page 679*

* not 512 or 768 constructed by Besche and Eick. ters, London Math. Soc. Monographs (N. S.), vol. 11, Among these, the 56092 groups of order 256 Clarendon Press, Oxford, 1995. were found by O’Brien, and it is expected that [9] Klaus Lux and Herbert Pahlings, Computational there are more than a million groups of order aspects of representation theory of finite groups (G. O. Michler and C. M. Ringel, eds.), Represen- 512. tation Theory of Finite Groups and Finite-Di- The 42038 transitive permutation groups of mensional Algebras, Progr. Math., vol. 95, degree at most 31, constructed by Hulpke, are Birkhauser-Verlag, 1991, pp. 37–64. also available. In principle this list can be used [10] J. NeubÜser, An elementary introduction to coset to compute Galois groups; the computations are table methods in computational group theory (C. M. practical up to degree 15. Nonaffine primitive Campbell and E. F. Robertson, eds.), Groups – St. permutation groups are listed up to degree 1000 Andrews 1981, London Math. Soc. Lecture Note by Dixon and Mortimer and solvable permuta- Ser., vol. 71, Cambridge Univ. Press, 1982, pp. tion groups up to degree 255 by Short. The list 1–45. of 4-dimensional space groups by Brown, Bülow, [11] ———, An invitation to computational group the- ory (C. M. Campbell, T. C. Hurley, E. F. Robertson, Neubüser, Wondratschek, and Zassenhaus and S. J. Tobin, and J. J. Ward, eds.), Groups ‘93 – Gal- conjugacy class representatives of irreducible way/St. Andrews, London Math. Soc. Lecture Note maximal finite subgroups of GLn(Q) for n 24 ≤ Ser., vol. 212, Cambridge Univ. Press, 1995, pp. by Nebe and Plesken are now also online. In the 457–475. section “Representation Theory” we already men- [12] Ákos Seress, Nearly linear time algorithms for per- tioned the availability of ordinary and modular mutation groups: An interplay between theory character tables. Probably just the sizes of these and practice, Acta Appl. Math. (1997), to appear. databases indicate the computational difficulties [13] Charles C. Sims, Computation with finitely pre- obtaining them. sented groups, Cambridge Univ. Press, 1994. The applications of CGT in group theory are too numerous to list here. We just mention one of them: the construction of some of the sporadic finite simple groups by Sims and Norton. Both GAP and Magma support the application of CGT to related fields such as other areas of abstract algebra, graph theory, and coding the- ory. Magma covers other areas of symbolic com- putation as well; it has a particularly strong number theory component. Finally, we mention two stand-alone programs that may be used to investigate finitely presented groups. QUOTPIC by Holt and Rees is a user-friendly graphical in- terface to display factor groups of fp-groups, while the system Magnus, in the beta-release stage, works with infinite fp-groups.*

*References [1] Michael D. Atkinson, ed., Computational group theory, Academic Press, London, New York, 1984; Durham, 1982. [2] John Cannon, ed., Computational group theory I, J. Symbolic Comput. 9 (5-6) (1990). [3] ——— , Computational group theory II, J. Symbolic Comput. 12 (4-5) (1991). [4] John Cannon and George Havas, Algorithms for groups, Australian Comput. J. 24 (1992), 51–60. [5] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Clarendon Press, Oxford, 1985. [6] Larry Finkelstein and William M. Kantor, eds., Groups and computation, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 11, Amer. Math. Soc., Providence, RI, 1993. [7] ——— , Groups and computation II, DIMACS Ser. Dis- crete Math. Theoret. Comput. Sci., vol. 28, Amer. Math. Soc., Providence, RI, 1997. [8] Christoph Jansen, Klaus Lux, Richard Parker, and Robert Wilson, An atlas of Brauer charac-*

*JUNE/JULY 1997 NOTICES OF THE AMS 679*