The algorithmic theory of polycyclic-by-finite groups. (English) Zbl 0774.20019

A polycyclic group is a group having a normal series with cyclic factors; a polycyclic-by-finite group is a group containing a polycyclic subgroup of finite index. Polycyclic groups were introduced by K. A. Hirsch in 1938 [Proc. Lond. Math. Soc., II. Ser. 44, 53–60 (1938; Zbl 0018.14505); 44, 336–344 (1938; Zbl 0019.15602; JFM 64.0066.01)]; an exposition of the basic theory of these groups can be found in D. Segal’s book [Polycyclic groups. Cambridge Tracts Math. 82. Cambridge etc.: Cambridge University Press (1983; Zbl 0516.20001)].
The paper under review is concerned with producing algorithms for answering various group-theoretic questions about polycyclic-by-finite groups. Some results (such as solutions to the generalized word problem and the conjugacy problem) have already been obtained by other authors; these results are discussed in detail in the introduction to the paper. The new results obtained in the paper are quite extensive. Taken together with the paper by D. Segal [Proc. Lond. Math. Soc., III. Ser. 61, 497–528 (1990; Zbl 0674.20020)] they provide a comprehensive treatment of algorithmic questions concerning polycyclic-by-finite groups.
There are far too many results to discuss in a short review, but statements of two key results should give a flavour of the material: (1) There is an algorithm which on being given a finite presentation of a polycyclic-by-finite group and a finite set \(X\) of words in the generators, produces a finite presentation of the subgroup generated by \(X\) (Theorem 3.4); (2) There is an algorithm which, on being given a finite subset \(Y\) of \(\mathrm{GL}(n,\mathbb{Z})\), decides if the subgroup generated by \(Y\) is polycyclic-by-finite, and, if so produces a finite presentation of it (Theorem 4.1). [In relation to (2), recall that every polycyclic-by- finite group can be embedded in \(\mathrm{GL}(n,\mathbb{Z})\) for some \(n\) (L. Auslander, Ann. Math. (2) 86, 112–116 (1967; Zbl 0149.26904); R. G. Swan, Proc. Am. Math. Soc. 18, 573–574 (1967; Zbl 0153.03801)].


20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20F05 Generators, relations, and presentations of groups
20F16 Solvable groups, supersolvable groups
20F19 Generalizations of solvable and nilpotent groups
20E07 Subgroup theorems; subgroup growth
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
Full Text: DOI


[1] Auslander, L, On a problem of philip Hall, Ann. of math. (2), 86, 112-116, (1967) · Zbl 0149.26904
[2] Baer, R, Überauflösbare gruppen, (), 11-28 · Zbl 0092.02004
[3] Baumslag, G; Cannonito, F.B; Miller, C.F, Infinitely generated subgroups of finitely presented groups, I, Math. Z., 153, 117-134, (1977) · Zbl 0332.20009
[4] Baumslag, G; Cannonito, F.B; Miller, C.F, Some recognizable properties of solvable groups, Math. Z., 178, 289-295, (1981) · Zbl 0455.20027
[5] Baumslag, G; Cannonito, F.B; Miller, C.F, Computable algebra and group embeddings, J. algebra, 69, 186-212, (1981) · Zbl 0497.20023
[6] Baumslag, G; Gildenhuys, D; Strebel, R, Algorithmically insoluble problems about finitely presented solvable groups, Lie and associative algebras, I, J. pure appl. algebra, 39, 53-94, (1986) · Zbl 0577.20021
[7] Bieri, R, Homological dimension of discrete groups, Queen mary college math. notes, (1976), London · Zbl 0357.20027
[8] Borevič, Z.I; Šafarevič, I.R, Number theory, (1966), Academic Press New York
[9] Curtis, C.W; Reiner, I, Representation theory of finite groups and associative algebras, (1966), Interscience New York
[10] Formanek, E, Conjugacy separability of polycyclic groups, J. algebra, 42, 1-10, (1976) · Zbl 0345.20031
[11] Gruenberg, K.W, Cohomological topics in group theory, () · Zbl 0205.32701
[12] Grunewald, F, Solution of the conjugacy problem in certain arithmetic groups, (), 101-139
[13] Grunewald, F; Segal, D, Conjugacy in polycyclic groups, Comm. algebra, 6, 775-798, (1978) · Zbl 0403.20025
[14] Hirsch, K.A, On infinite soluble groups, I, (), 53-60 · Zbl 0018.14505
[15] Hirsch, K.A, On infinite soluble groups, III, (), 184-194 · Zbl 0063.02021
[16] Hirsch, K.A, On infinite soluble groups, IV, J. London math. soc., 21, 81-85, (1952) · Zbl 0046.02003
[17] Hirsch, K.A, On infinite soluble groups, V, J. London math. soc., 29, 250-251, (1954) · Zbl 0055.01603
[18] Kegel, O.H, Über den normalisator von subnormalen und erreichbaren untergruppen, Math. ann., 163, 248-258, (1966) · Zbl 0135.04804
[19] Kharlampovič, O.G, A finitely presented soluble group with insoluble word problem, Izv. akad. nauk. ser. mat., 45, 852-873, (1981) · Zbl 0485.20023
[20] Lennox, J.C, A note on quasinormal subgroups of finitely generated groups, J. London math. soc. (2), 24, 127-128, (1981) · Zbl 0526.20020
[21] Lennox, J.C; Wilson, J.S, On products of subgroups in polycyclic groups, Arch. math. (basel), 33, 305-309, (1979) · Zbl 0426.20026
[22] Magnus, W; Karrass, A; Solitar, D, Combinatorial group theory, (1966), Interscience New York
[23] Mal’cev, A.J, On certain classes of infinite soluble groups, Mat. sb., 28, 567-588, (1951)
[24] Mal’cev, A.J, Homomorphisms onto finite groups, Ivanov. GoS. ped. inst. Učen. zap., 18, 49-60, (1958)
[25] Remeslennikov, V.N, Conjugacy in polycyclic groups, Algebra i logika, 8, 712-725, (1969)
[26] Remeslennikov, V.N, An algorithmic problem for nilpotent groups and rings, Sibirsk. mat. Z̆h., 20, 1077-1081, (1979) · Zbl 0426.20024
[27] Rhemtulla, A, A minimality property of polycyclic groups, J. London math. soc., 42, 456-462, (1967) · Zbl 0166.01804
[28] Robinson, D.J.S, Splitting theorems for infinite groups, (), 441-470
[29] {\scD. J. S. Robinson}, Algorithmic problems for automorphisms and endomorphisms of infinite soluble groups, preprint.
[30] Segal, D, Polycyclic groups, (1983), Cambridge · Zbl 0516.20001
[31] Segal, D, Decidable properties of polycyclic groups, (), 497-528 · Zbl 0674.20020
[32] Seidenberg, A, Constructions in a polynomial ring over the ring of integers, Amer. J. math., 100, 685-703, (1978) · Zbl 0416.13013
[33] Stonehewer, S.E, Permutable subgroups of infinite groups, Math. Z., 125, 1-16, (1972) · Zbl 0219.20021
[34] Swan, R.G, Representations of polycyclic groups, (), 573-574 · Zbl 0153.03801
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