zbMATH — the first resource for mathematics

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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.