Algorithmic Problems for Solvable Groups

S.I. Adian, W.W. Boone, G . Higman, eds., Word Problems I1 @ North-Holland Publishing Company (1980) 337-346

ALGORITHMIC PROBLEMS FOR SOLVABLE GROUPS V.N. REMESLENNIKOV and N.S. ROMANOVSKII Institute of Mathematics, Nooosibirsk

$1. Introduction

Classical algorithmic problems - t h e word problem, conjugacy problem, and isomorphism problem - arose in the theory of groups from topolocal considerations and were formulated for t h e first time at the beginning of this century by Dehn. In the middle fifties it was shown that these problems have negative solutions in the class of all groups. However, for important classes of groups such as nilpotent and solvable groups, these problems remained unsolved. At t h e present moment a number of interesting results have been obtained in this field, and this paper is intended to survey them. Let us recall the main definitions and t h e formulation of t h e classical algorithmic problems for t h e class of all groups. We call the pair ( X , R ) t h e presentation of group G, where X is a non-empty set and R is a set of t h e words in t h e alphabet X U X - ' . The group G is a factor group of the free group with the base X by the normal subgroup, generated by the set R . With this understanding we shall write G = (X, R ) ; further, X is called the set of the generators of group G, and R - the set of defining relators. If t h e sets X and R are finite, we say that group G is finitely presented. We shall also consider the case when X is finite and R is recursively enumerable. In this case G is called a recursively presented group. Let G be a given finitely presented (recursively presented) group. The word problem, conjugacy problem and membership problem are the problems of establishing the existence of algorithms to decide respectively (i) whether or not an arbitrary element (word) of group G equals 1; (ii) whether or not two arbitrary elements from G are conjugate in G ; (iii) for a given arbitrary finite set { u , u , , . . . ., u,} of elements of G 337

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whether or not the element u belongs to the subgroup generated by UI,. . . , V".

Finally, let = {GiI i E N } be a recursive class of finitely presented groups. The isomorphism problem for the class C#I consists in establishing the existence of an algorithm to decide for arbitrary i, j whether or not the groups G, and G, are isomorphic. P.S. Novikov [lo] and W.W. Boone [27] constructed examples of finitely presented groups, for which the word problem was solved negatively. This implied the negative solution of the conjugacy and membership problems. Aided by this result, S.I. Adian [l] proved that the isomorphism problem in the class of all groups also has a negative solution. In addition, algorithmic unsolvability was proved for a number of other problems, such as the recognition of the following properties: nilpotency, finiteness, simplicity or unity of a given group presentation. Let H be a subgroup of the finitely presented group G, generated by the elements u I , . .. , u r nThe . set of all the words from the u, is recursively enumerable, as is the set of all the words from G equal to 1. The intersection of these two sets forms the recursively enumerable set of the presenting relations of group H in the generators u I , .. . ,u,. Therefore, the finitely generated subgroup of the finitely presented group is recursively presented. H igman (301 showed that the converse statement is also correct; i.e. every recursively presented group is embeddable in some finitely presented group.

92. The setting of the problems

Thus we see that the majority of algorithmic problems has a negative solution in the class of all groups. Therefore, it is natural to study algorithmical problems with additional restrictions on t h e groups considered. In a number of papers, such restrictions were on the form of the defining relations, e.g., Magnus' theorem [32], proved in 1932, and which deals with the solvability of the word problem for groups with one defining relation. However, we shall consider restrictions having a natural group-theoretic character, such as nilpotency, polycyclicity and solvability. Here, two approaches to such problems are possible. The first approach consists in the algorithmic problems being considered for a finitely presented group (or a set of groups) under the assumption that such group turns out to be, say, solvable of given steps of solvability;

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i.e. it is assumed that t h e condition of solvability results from the defining relators. In t h e second approach, groups which are finitely presented in a given variety are studied. This means t h e following: Let V be a variety given by a finite set of identities. We say that a group G is given in the variety V by generators X and relations R, if it is a factor group of t h e free group of the variety V with the base X by the normal subgroup, generated by the set R. Finitely presented group and recursively presented group in the variety V are defined in the obvious way. Let us introduce the corresponding designations FP, RP, FPV, RPV for the classes: finitely presented, recursively presented, finitely presented in the variety V, recursively presented in the variety V. In t h e language of these classes t h e first of the formulated approaches leads to the study of groups from F P n V, if the additional condition consists of membership in t h e variety V. In the second approach groups from FPV are studied. Evident inclusions take place:

F P n V c F P V C R P n V=RPV. For some varieties these relations may be made more exact. If V is t h e variety of groups of nilpotency class c, then it is well known that FP f l V = FPV = RPV. where 1 3 2 For the variety of 1-step solvable groups, denoted by d’, strict inclusion takes place: F P n V 3 the inclusion is strict FPV < RPV, as it was shown by A.L. Shmelkin [23], who proved that the free metabelian group is not finitely presented in d’. Recalling Higman’s theorem about the embedding of recursively presented groups into finitely presented ones, we may formulate a few which acquire an important meaning, problems for the varieties d’, taking into account V.N. Remeslennikov’s theorem, which we shall deal with below.

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Problem 1. Can any RPd'-group, where I >3, be embedded into an FPdm-groupfor suitable m ? Problem 2. Can any FPdI-group, where I >3, be embedded into an FP r l d"-group for suitable m ? Problem 3. Can any RPd-group, where I > 3 , be embedded into an FP rl A -group for suitable m ? For I = 2 these problems have been solved positively, since G. Baumslag [25] and V.N. Remeslennikov [13, 151 have shown that any finitely generated metabelian group can be embedded into a finitely presented metabelian group. 63. Classical algorithmic problems for solvable groups Before supplying the formulations of the main results obtained, we shall take note of the connection of the algorithmic problems with the finite approximatability of the groups. Let T be some group-theoretic predicate. We say that the group G is finitely approximated with respect to the predicate T , if for any collection of elements which are not related under T, there exists a homomorphism of group G to a finite group H such that the images of the elements considered are not related under T in H. Let us consider, for instance, the predicate: to be equal to the element 1. Groups which are finitely approximated with respect to this predicate are called residually finite. Let G be a group finitely presented in the variety V. Let the variety V be given by a finite set of identities. If G is residually finite, then the word problem can be positively solved in G. I.e., the set of words representing 1 is recursive. It is evident that this set is recursively enumerable. Therefore, it is enough to show that its complement is also recursively enumerable. Actually, because of the residual finiteness the complement may be enumerated by searching through all possible homomorphisms of group G to finite groups. Similarly, if the group G is residually finite with respect to conjugacy or membership, then in it the problems of conjugacy and membership are solved positively. Let us consider now the word problem for groups in d'.By Hall's theorem [29] finitely generated metabelian groups are residually finite, therefore, the word problem for groups finitely presented in the variety d 2 ,is solved positively. On the other hand, V.N. Remeslennikov's

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theorem [14] states that for 1 > 5 there exists a group finitely presented irl the variety d’, for which the word problem is solved negatively. To be more precise, for each I a specific representation of this group can be obtained. Since FP n d’< F P d ’ for 1 > 3, we are led to the following question:

Problem 4. Does there exist a solvable finitely presented group with unsolvable word problem? Let us note that a similar question was formulated by S.I. Adian, who asked if there exists a group with unsolvable word problem which satisfies some nontrivial identity. It is clear that an affirmative answer to problem 2 and 3 gives an affirmative answer to problem 4. The cases 1 = 3, 4 are not covered by V.N. Remeslennikov’s theorem. Hence, we have the following question:

Problem 5. Is the word problem solvable for groups from F P d 3 or FPd4? As for the conjugacy problem, we know still less about it. It follows from V.N. Remeslennikov’s theorem that for 1 > 5 there exist examples of groups finitely presented in the variety d’, for which the conjugacy problem has negative solution. However, this problem remains uninvestigated for solvable groups of 2, 3 and 4 steps. It should be noted that it cannot be solved by using the finite approximation method, because M.I. Kargapolov and E.I. Timoshenko [5], and Wehrfritz [35],have constructed examples of finitely generated metabelian groups which are not residually finite with respect to conjugacy. As for the membership problem, there exist two main results. On the one hand, N.S. Romanovskii [18] proved that it is solved positively for metabelian groups. On the other hand V.N. Remeslennikov [14], for 1 3 4 , has constructed an example of a group which is finitely presented in the variety d‘,and with a finitely generated subgroup for which the membership problem is solved negatively. Hence, we have the following question:

Problem 6. Is the membership problem solvable for groups in F P d 3 ? By using V.N. Remeslennikov’s theorem, for each I2 7, there was for which there constructed in [6] a group G finitely presented in d’,

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exists no algorithm, deciding for any group H finitely presented in d' whether or not it is isomorphic to G. By this example the isomorphism problem for F P d ' for 1 3 7 is solved negatively. Hence, we have the following

Problem 7. Is the isomorphism problem solvable for metabelian groups? We recall that in the class of all groups the isomorphism problem for a unit group is solved negatively. The analogous statement is wrong for the variety d', because there G # 1 if and only if G / G ' # 1. Therefore, the problem is reduced to the same problem for abelian groups, where, as is well known, it has a positive solution. We may also remark that for groups from F P d ' there exists an algorithm to decide whether a given group is finite or not, and in the positive case to find its multiplication table. This can be proved by induction on the solvability step of the group. It should be noted that, for solvable groups, there does not exist a general theorem on the algorithmic undecidability of group-theoretic properties, similar to the theorem of Adian-Rabin. Perhaps, we can prove such a theorem if we can solve the following:

Problem 8. Is there an algorithm which decides if groups in d', 1 > 2, are nilpotent? Now a few words about the methods of proof of the majority of theorems of this kind. They use the so-called Magnus embedding of a free solvable group, the essence of which lies in the following. Let 5 be a free solvable group of step 1 and rank n and T the left free module of rank n over the ring Z[F,]. Then the free solvable group Fl+Iof the step 1 + 1 and rank n can be embedded into the group of matrices M of the type (6 :), where f E R and t E T.We may consider the group M as the wreath product of the free abelian group of rank n with R. The Magnus embedding turns out to be convenient, for instance, in the case that N is a normal subgroup of E + , ;for in this case the embedding may be extended to the normal of group M,which represents the semidirect product of subgroup the subgroups consisting of unitriangular and diagonal matrices such n F,+,= "181. that Thanks to the Magnus embedding many questions concerning solvable groups can be replaced by corresponding questions about modules over group rings of lesser solvability step. In particular, a

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number of algebraic problems for two-step solvable groups become problems for free modules over the ring Z[%',,. . . ,Zn]. 94. Free solvable groups

As has been mentioned above, free solvable groups of finite rank are recursively defined in the class of all groups, and evidently finitely defined in the corresponding variety YI'. By the use of the Magnus embedding for a free solvable group, the positive solution of the word problem is easily established. Matteus [33] showed that in the wreath product of groups A and B the conjugacy problem is solved positively, if it is solved positively in A and B, and additionally, the membership problem for cyclic subgroups is positively solvable in B. Making use of this fact and of the Magnus embedding, M.I. Kargapolov and V.N. Remeslennikov [3] positively solved the conjugacy problem for free solvable groups by induction on the solvability step. Later on, V.N. Remeslennikov and V.S. Sokolov [16] established the finite approximatability of those groups with respect to conjugacy. R.A. Sarkisjan [19] proved that the conjugacy problem is positively solved for free polynilpotent groups. The next problem is still open. Problem 9. Is the membership problem solvable for free solvable groups? This problem is closed connected with the following. Problem 10. Is there an algorithm for solving finite systems of linear equations over Z[F,]? Here, F. is a free solvable group with solvability length 1. Let us also note that in an absolutely free group there exist algorithms (for instance, Nilsen's method), to decide if a given finite set is a basis. A.F. Krasnikov (7) has found a similar algorithm for free solvable groups. The class of groups with one defining relation in the variety a' is the closest to free solvable groups. We know little about these groups so far. Let us still note that N.S. Romanovskii [17] proved the analog of Magnus' Freiheitssatz for a group with one relation in the variety a', and the word problem with some restrictions on the single relation has been solved [18]. However, the general case remains open:

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Problem 11. Is the word problem solvable for groups with one defining relation in the variety ‘ill’1,3 3? 05. Algorithmic problems for nilpotent groups

In 1955 A.I. Mal’cev [8] showed that the word problem can be solved positively for nilpotent groups. He also showed [9], that finitely generated nilpotent groups are residually finite with respect to membership, hence the membership problem is solvable. Blackburn [26] has positively solved the conjugacy problem for nilpotent groups by the same method. M.I. Kargapolov and his students [4] raised and solved the following algorithmic problems for $3-power groups over a those of word, conjugacy, membership, separating of “good” ring ‘0-periodical part, intersection, the description of subgroups in the terms of relations. Thus, we see that in nilpotent groups the picture is diametrically opposed to that in the class of all groups we have the following:

a:

Problem 12. Is the isomorphism problem solvable for nilpotent groups? This problem cannot be solved by the method of finite approximation, for at present we know many examples of nonisomorphic finitely generated nilpotent groups with the same families of finite homomorphic images. Concerning this we should note Pickel’s theorem [34]. H e showed that there can only exist a finite number of groups with the same family of finite homomorphic images in the class of finitely generated nilpotent groups. It is highly probable that problem 12 will be answered negatively, hence, we have another interesting problem: Problem 13. Is the isomorphism problem solvable for groups with one defining relation in the variety of nilpotent groups with given nilpotency class? Such an algorithm exists for class 2-nilpotent groups [20]. 06. Algorithmic problems for polycyclic groups

It should be noted at once that any polycyclic group is finitely presented, so it is possible to discuss algorithmic problems for

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polycyclic groups. All results known to us here were obtained by means of finite approximatability. Initially, Hirsch [31] showed that polycyclic groups are residually finite. Then A.I. Mal’cev [9] showed that they are finitely approximated with respect to membership. Finally, V.N. Remeslennikov [12] and Formanek [28] proved the finite approximatability of polycyclic groups with respect to conjugacy. The isomorphism problem for polycyclic groups is still unsolved.

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the varieties of solvable and nilpotent groups of given steps, Matematiceskij sbornik 89, No. 1 (1972) 93-99 (Russian). [18] N.S. Romanovskii, On some algorithmic problems for solvable groups, Algebra i logika 13, No. 1 (1974) 2 6 3 4 (Russian). [19] R.A. Sarkisjan, Conjugacy in free polynilpotent groups, Algebra i logika 11, No. 6 (1972) 694-710 (Russian). [20] N.F. Sesekin, On the classification of metabelian (two-step nilpotent) torsion free groups, Ucenye zapiski Ural’skogo universiteta 1 9 (1965) 27-41 (Russian). [21] V.G. Sokolov, An algorithm for the word problem for a class of solvable groups, Sibirskij matematiceskij iurnal 12, No. 6 (1971) 140.5-1410 (Russian). (221 E.I. Timoshenko, Some algorithmic problems for metabelian groups, Algebra i logika 12, No. 2 (1973) 232-240 (Russian). [23] A.L. Shmel’kin, Wreaih products and varieties of groups, Izv. AN SSSR, ser. mat. 29 (1965) 149-170 (Russian). [24] G. Baumslag, Wreath products and finitely presented groups, Math. Z. 7.5 (1961) 22-28. [2.5] G. Baumslag, Subgroups of finitely presented metabelian groups, J. Austral. Math. Soc. 16 (1973) 98-110. [26] N. Blackburn, Conjugacy in nilpotent groups, Proc. Amer. Math. Soc. 16 (1965) 143- 148. [27] W. Boone, The word problem, Ann. Math. 70, No. 2 (1959) 207-265. [28] E. Formanek, Conjugate separability in polycyclic groups, J. Algebra 42, No. 1 (1976) 1-10, [29] P. Hall, On the finiteness of certain solvable groups, Proc. London Math. Soc. 9 (1959) 595-622. [30] G. Higman, Subgroups o n finitely presented groups, Proc. Roy. Soc. London A262 (1961) 455-475. (311 K.A. Hirsch, On infinite solvable groups, IV, J. London Math. Soc. 27 (1952) 81-85. 1321 W. Magnus, Das Identitats problem fur Gruppen mit einer definierenden Relation, Math. Ann. 106 (1932) 295-307. [33] J. Matteus, The conjugacy problem in wreath products and free metabelian groups, Trans. Amer. SOC. 121 (1966) 329-339. [34] P.F. Pickel, Finitely generated nilpotent groups with isomorphic finite quotients, Trans. Amer. Math. Soc. 160 (1971) 327-341. [35] Wehrfritz, Two examples of solvable groups that are not conjugacy separable, J. London Math. Soc. (2). 7 (1973) 312-316.