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An Introduction to Computational 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 . Also, nu- the graphs on Figure 1 isomorphic? merous stand-alone programs and smaller sys- What is the of the poly- tems are available. Cnomial x8 +2x7 +28x6 + 1728x + 3456? What GAP can be obtained by anonymous ftp from are the possible groups of ? 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 . 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 of generators: for example, 50 bits are to the most frequently used representations of enough to define GL5(2), a group of groups: groups, 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, . 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 . 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 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 group methods was rwth-aachen.de/˜Akos.Seress/. E. O’Brien’s written by J. Neubüser. database of papers on group theory, including

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Finitely Presented Groups Let G = E be a presentation for a h |Ri group G : E = g1,... ,gn is a of { } generators, and = r1 =1,... ,rm =1 is R { } a set of defining relations. Each ri is a , 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 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 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 , 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 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 , 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, . 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 manipulation and the com- 1 2 t j defined, is the of the coset kgi gi . putation of 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. , arose, and the development of Finally, for each hj = gj1 gjt of the first integrated system, the Aachen-Sydney H, we maintain a 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 , 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-

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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 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 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 =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 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 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 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 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 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 about 100. example, as permutation or matrix groups, but

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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 . 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

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first step, an 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 . group. Suppose that an epimorphism ϕ is known The basic ideas of permutation group ma- onto a nilpotent or 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 B =(β1,... ,βm) of ≤ G onto an 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 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 ; 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 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 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 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