Decision Problems for Groups — Survey and Reflections

  • Charles F. MillerIII
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 23)

Abstract

This is a survey of decision problems for groups, that is of algorithms for answering various questions about groups and their elements. The general objective of this area can be formulated as follows:
  • Objective: To determine the existence and nature of algorithms which decide
    • local properties — whether or not elements of a group have certain properties or relationships;

    • global properties— whether or not groups as a whole possess certain properties or relationships.

Keywords

Expense Resid Univer 

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References

  1. [1]
    S. I. Adian, Algorithmic unsolvability of problems of recognition of certain properties of groups, Dokl. Akad. Nauk SSSR 103, 533–535 (1955).MATHGoogle Scholar
  2. [2]
    S. I. Adian, The unsolvability of certain algorithmic problems in the theory of groups, Trudy Moskov. Mat. Obsc. 6, 231–298 (1957).MATHGoogle Scholar
  3. [3]
    S. I. Adian, Finitely presented groups and algorithms, Dokl. Akad. Nauk SSSR 117, 9–12 (1957).MATHGoogle Scholar
  4. [4]
    J. Alonso, Combings of groups,this volume.Google Scholar
  5. [5]
    J. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short Notes on negatively curved groups, MSRI preprint, 1989; to appear in the proceedings of the conference “Group theory from a geometrical viewpoint” held at Trieste, 1990.Google Scholar
  6. [6]
    D. J. Anick, On the homology of associative algebras, Trans. Amer. Math. Soc. 296, 641–659 (1986).CrossRefMathSciNetGoogle Scholar
  7. [7]
    A. V. Anisimov, Groups languages, Kybernetika 4, 18–24 (1971).Google Scholar
  8. [8]
    W. Ballmann, E. Ghys, A. Haefliger, “ Sur les groupes hyperboliques d’après Mikhael Gromov” (Notes of a seminar held at Berne), edited by E. Ghys and P. de la Harpe, Birkhäuser, Progress in Mathematics Series, 1990.Google Scholar
  9. [9]
    G. Baumslag, Finitely generated residually torsion free nilpotent groups,in preparation.Google Scholar
  10. [10]
    G. Baumslag, W. W. Boone and B. H. Neumann, Some unsolvable problems about elements and subgroups of groups, Math. Scand. 7, 191–201 (1959).MATHMathSciNetGoogle Scholar
  11. [11]
    G. Baumslag, F. B. Cannonito and C. F. Miller III, Computable algebra and group embeddings, Journ. of Algebra 69, 186–212 (1981).CrossRefMATHMathSciNetGoogle Scholar
  12. [12]
    G. Baumslag, F. B. Cannonito and C. F. Miller III, Infinitely generated subgroups of finitely presented groups, I, Math. Zeit. 153, 117–134 (1977).CrossRefGoogle Scholar
  13. [13]
    G. Baumslag, F. B. Cannonito and C. F. Miller III, Infinitely generated subgroups of finitely presented groups, II, Math. Zeit. 172, 97–105 (1980).CrossRefGoogle Scholar
  14. [14]
    G. Baumslag, F. B. Cannonito and C. F. Miller III, Some recognizable properties of solvable groups, Math. Zeit. 178, 289–295 (1981).CrossRefGoogle Scholar
  15. [15]
    G. Baumslag, F. B. Cannonito, D. J. S. Robinson and D. Segal, The algorithmic theory of polycyclic-by-finite groups, to appear in Journal of Algebra.Google Scholar
  16. [16]
    G. Baumslag, E. Dyer and C. F. Miller III, On the integral homology of finitely presented groups, Topology 22, 27–46 (1983).CrossRefMATHMathSciNetGoogle Scholar
  17. [17]
    G. Baumslag, S. M. Gersten, M. Shapiro and H. Short, Automatic groups and amalgams, to appear. (A summary appears in this volume.)Google Scholar
  18. [18]
    G. Baumslag, D. Gildenhuys and R. Strebel, Algorithmically insoluble problems about finitely presented soluble groups, Lie and associative algebras, I, Journ. Pure and Appl. Algebra 39, 53–94 (1986).MATHMathSciNetGoogle Scholar
  19. [19]
    G. Baumslag and J. E. Roseblade, Subgroups of direct products of free groups, Journ. London Math. Soc. (2) 30, 44–52 (1984).CrossRefMATHMathSciNetGoogle Scholar
  20. [20]
    G. Baumslag and D. Solitar, Some two-generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. 68, 199–201 (1962).CrossRefMATHMathSciNetGoogle Scholar
  21. [21]
    W. W. Boone, Certain simple unsolvable problems in group theory, I, II, III, IV, V, VI, Nederl. Akad. Wetensch Proc. Ser. A57, 231–237,492–497 (1954), 58, 252–256,571–577 (1955), 60, 22–27, 227–232 (1957).Google Scholar
  22. [22]
    W. W. Boone Word problems and recursively enumerable degrees of unsolvability. A sequel on finitely presented groups., Annals of Math. 84, 49–84 (1966).MATHMathSciNetGoogle Scholar
  23. [23]
    W. W. Boone and G. Higman, An algebraic characterization of the solvability of the word problem, J. Austral. Math. Soc. 18, 41–53 (1974).CrossRefMATHMathSciNetGoogle Scholar
  24. [24]
    W. W. Boone and H. Rogers Jr., On a problem of J.H.C. Whitehead and a problem of Alonzo Church, Math. Scand. 19, 185–192 (1966).MATHMathSciNetGoogle Scholar
  25. [25]
    K. S. Brown, The geometry of rewriting systems: a proof of the Anick-GrovesSquier theorem, this volume.Google Scholar
  26. [26]
    J. W. Cannon, D. B. A. Epstein, D. F. Holt, M. S. Patterson and W. P. Thurston, Word processing and group theory, preprint, University of Warwick, 1988.Google Scholar
  27. [27]
    C.R.J. Clapham, Finitely presented groups with word problems of arbitrary degrees of insolubility, Proc. London Math. Soc. (3) 14, 633–676 (1964).CrossRefMATHGoogle Scholar
  28. [28]
    C.R.J. Clapham, An embedding theorem for finitely generated groups, Proc. London Math. Soc. (3) 17, 419–430 (1967).CrossRefMATHGoogle Scholar
  29. [29]
    D. J. Collins, Representation of Turing reducibility by word and conjugacy problems in finitely presented groups, Acta Mathematica 128, 73–90 (1972).CrossRefMathSciNetGoogle Scholar
  30. [30]
    D. J. Collins and C. F. Miller III, The conjugacy problem and subgroups of finite index, Proc. London Math. Soc.ser 3 34, 535–556 (1977).Google Scholar
  31. [31]
    M. Dehn, Uber unendliche diskontinuerliche Gruppen, Math. Ann. 69, 116–144 (1911).Google Scholar
  32. [32]
    M. J. Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81, 449–457 (1985).MATHGoogle Scholar
  33. [33]
    V. H. Dyson The word problem and residually finite groups, Notices Amer. Math. Soc. 11, 734 (1964).Google Scholar
  34. [34]
    E. Formanek, Conjugacy separability in polycyclic groups, Journ. Algebra 42, 1–10 (1976).CrossRefMathSciNetGoogle Scholar
  35. [35]
    A. A. Fridman, Degrees of unsolvability of the word problem for finitely defined groups, Izdalel’stvo “Nauk”, Mpscow, 193pp (1967).Google Scholar
  36. [36]
    S. M. Gersten, Dehn functions and ll-norms of finite presentations,this volume.Google Scholar
  37. [37]
    S. M. Gersten and H. Short, Small cancellation theory and automatic groups, Inventiones Math. 102, 305–334 (1990).CrossRefMATHMathSciNetGoogle Scholar
  38. [38]
    S. M. Gersten and H. Short, Small cancellation theory and automatic groups, Part II,to appear in Inventiones Math.Google Scholar
  39. [39]
    S. M. Gersten and H. Short, Rational subgroups of biautomatic groups,to appear in Annals of Math.Google Scholar
  40. [40]
    C. M. Gordon, Some embedding theorems and undecidability questions for groups, unpublished manuscript, 1980.Google Scholar
  41. [41]
    A. V. Gorjaga and A. S. Kirkinskii, The decidability of the conjugacy problem cannot be transferred to finite extensions of groups, Algebra i Logica 14, 393–406 (1975).MathSciNetGoogle Scholar
  42. [42]
    A. P. Goryushkin, Imbedding of countable groups in 2-generator groups, Mat. Zametki 16, 231–235 (1974).MATHGoogle Scholar
  43. [43]
    M. D. Greendlinger, Dehn’s algorithm for the word problem, Comm. Pure Appl. Math. 13, 67–83 (1960).Google Scholar
  44. [44]
    M. D. Greendlinger, On Dehn’s algorithms for the conjugacy and word problems with applications, Comm. Pure Appl. Math. 13, 641–677 (1960).MATHMathSciNetGoogle Scholar
  45. [45]
    M. Gromov, Hyperbolic groups, in “Essays on group theory”, MSRI series vol. 8, edited by S. Gersten, Springer-Verlag, 1987Google Scholar
  46. [46]
    J. R. J. Groves, Rewriting systems and homology of groups,to appear in Proceedings of Canberra Group Theory Conf. ed. L. G. Kovacs,in Lecture Notes in Math. (Springer).Google Scholar
  47. [47]
    J. R. J. Groves and G. C. Smith, Rewriting systems and soluble groups, University of Melbourne preprint, 1989.Google Scholar
  48. [48]
    F. Grunewald and D. Segal, Some general algorithms. I: Arithmetic groups, II: Nilpotent groups, Annals of Math. 112 531–583, 585–617 (1980).Google Scholar
  49. [49]
    F. Grunewald and D. Segal, Decision problems concerning S-arithmetic groups, Journ. Symbolic Logic 90, 743–772 (1985).CrossRefMathSciNetGoogle Scholar
  50. [50]
    M. Hall Jr., “The theory of groups”, Macmillan, New York, 1959.MATHGoogle Scholar
  51. [51]
    P. Hall, Finiteness conditions for soluble groups, Proc. London Math. Soc. III4, 419–436 (1954).Google Scholar
  52. [52]
    P. Hall, Embedding a group in a join of given groups, J. Austral. Math. Soc. 17, 434–495 (1974).CrossRefMATHMathSciNetGoogle Scholar
  53. [53]
    G. Higman, A finitely generated infinite simple group, J. London Math. Soc. 26, 61–64 (1951).CrossRefMATHMathSciNetGoogle Scholar
  54. [54]
    G. Higman, Subgroups of finitely presented groups, Proc. Royal Soc. London Ser. A 262, 455–475 (1961).CrossRefMATHGoogle Scholar
  55. [55]
    G. Higman, B. H. Neumann and H. Neumann, Embedding theorems for groups, J. London Math. Soc. 24, 247–254 (1949).CrossRefMathSciNetGoogle Scholar
  56. [56]
    J. Hoperoft and J. Ullman, “Introduction to automata theory, languages and computation”, Addison-Wesley, Boston, 1979.Google Scholar
  57. [57]
    A. Juhasz, Solution of the conjugacy problem in one relator groups,this volume.Google Scholar
  58. [58]
    O. Kharlampovich, A finitely presented soluble group with insoluble word problem, Izvestia Akad. Nauk Ser. Mat. 45, 852–873 (1981).MATHMathSciNetGoogle Scholar
  59. [59]
    Yu. G. Kleiman, Identities and some algorithmic problems in groups, Dokl. Akad. Nauk SSSR 244, 814–818 (1979).Google Scholar
  60. [60]
    Yu. G. Kleiman, On some questions of the theory of varieties of groups, Dokl. Akad. Nauk SSSR 257, 1056–1059 (1981).Google Scholar
  61. [61]
    Yu. G. Kleiman, On identities in groups, Trudy Moskov Mat. Obshch. 44, 62–108 (1982).MATHGoogle Scholar
  62. [62]
    Yu. G. Kleiman, Some questions of the theory of varieties of groups, Izv. Akad. Nauk SSSR Ser. Mat. 47, 37–74 (1983).MATHMathSciNetGoogle Scholar
  63. [63]
    A. V. Kuznetsov, Algorithms as operations in algebraic systems, Izv. Akad. Nauk SSSR Ser Mat (1958).Google Scholar
  64. [64]
    P. Le Chenadec, “Canonical forms in finitely presented algebras”, Res. Notes in Theoret. Comp. Sci., Pitman (London-Boston) and Wiley(New York ), 1986.Google Scholar
  65. [65]
    R. C. Lyndon, On Dehn’s algorithm, Math. Ann. 166, 208–228 (1966).CrossRefMATHMathSciNetGoogle Scholar
  66. [66]
    R. C. Lyndon and P. E. Schupp, “Combinatorial Group Theory”, Springer-Verlag, Berlin-Heidleberg-New York,1977.Google Scholar
  67. [67]
    I. G. Lysbnok, On some algorithmic properties of hyperbolic groups, Izv. Akad. Nauk SSSR Ser. Mat. 53 No. 4 (1989); English transl. in Math. USSR Izv. 35 145–163 (1990).Google Scholar
  68. [68]
    K. Madlener and F. Otto, About the descriptive power of certain classes of string-rewriting systems, Theoret. Comp. Sci. 67 143–172 (1989).CrossRefMATHMathSciNetGoogle Scholar
  69. [69]
    W. Magnus, Über diskontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitsats), J. Reine Angew. Math. 163, 141–165 (1930).MATHGoogle Scholar
  70. [70]
    W. Magnus, Das Identitätsproblem für Gruppen mit einer definierenden Relation, Math. Ann.106, 295–307 (1932).Google Scholar
  71. [71]
    W. Magnus, A. Karrass and D. Solitar, “Combinatorial Group Theory”, Wiley, New York, 1966.MATHGoogle Scholar
  72. [72]
    A. I. Malcev, On isomorphic matrix representations of infinite groups,Mat. Sb. 8 405–422 (1940).Google Scholar
  73. [73]
    A. I. Malcev, Homomorphisms onto finite groups, Ivanov. Gos. Ped. Inst. Ucen. Zap. 18, 49–60 (1958).Google Scholar
  74. [74]
    J. C. C. McKinsey, The decision problem for some classes of sentences without quantifiers, Journ. Symbolic Logic 8, 61–76 (1943).CrossRefMATHMathSciNetGoogle Scholar
  75. [75]
    K. A. Mihailova, The occurrence problem for direct products of groups, Dokl. Akad. Nauk SSSR 119, 1103–1105 (1958).MATHGoogle Scholar
  76. [76]
    K. A. Mihailova, The occurrence problem for free products of groups, Dokl. Akad. Nauk SSSR 127, 746–748 (1959).MATHGoogle Scholar
  77. [77]
    C. F. Miller III, “On group theoretic decision problems and their classification”, Annals of Math. Study 68, Princeton University Press, Princeton, NJ, 1971.Google Scholar
  78. [78]
    C. F. Miller III, The word problem in quotients of a group, in “Aspects of Effective Algebra”, ed. J.N. Crossley, Proceedings of a conference at Monash University Aug. 1979, Upside Down A Book Company, Steel’s Creek, (1981), 246–250.Google Scholar
  79. [79]
    C. F. Miller III and P. E. Schupp, Embeddings into hopfian groups, Journ. of Algebra 17, 171–176 (1971).CrossRefMATHMathSciNetGoogle Scholar
  80. [80]
    A. W. Mostowski, On the decidability of some problems in special classes of groups, Fund. Math. 59, 123–135 (1966).MATHGoogle Scholar
  81. [81]
    D. E. Muller and P. E. Schupp, Groups, the theory of ends and context free languages, J. Comput. and Sys. Sci. 26, 295–310 (1983).CrossRefMATHMathSciNetGoogle Scholar
  82. [82]
    D. E. Muller and P. E. Schupp, The theory of ends, pushdown automata, and second-order logic, Theoret. Comput. Sci. 37, 51–75 (1985).MATHMathSciNetGoogle Scholar
  83. [83]
    B. H. Neumann, Some remarks on infinite groups, J. London Math. Soc. 12, 120–127 (1937).Google Scholar
  84. [84]
    B. B. Newman, Some results on one relator groups, Bull. Amer. Math. Soc 74, 568–571 (1968).CrossRefMATHMathSciNetGoogle Scholar
  85. [85]
    G. A. Noskov, On conjugacy in metabelian groups, Mat. Zametki 31 495–507 (1982).MATHGoogle Scholar
  86. [86]
    P. S. Novikov On the algorithmic unsolvability of the problem of identity, Dokl. Akad. Nauk SSSR 85, 709–712 (1952).MATHGoogle Scholar
  87. [87]
    P. S. Novikov On the algorithmic unsolvability of the word problem in group theory, Trudy Mat. Inst. Steklov 44, 1–143 (1955).Google Scholar
  88. [88]
    M. O. Rabin, Recursive unsolvability of group theoretic problems,Annals of Math. 67 172–194(1958).Google Scholar
  89. [89]
    M. O. Rabin, Computable algebra, general theory and theory of computable fields,Trans. Amer. Math. Soc. 95 341–360(1960).Google Scholar
  90. [90]
    V. N. Remmeslennikov, Finite approximability of metabelian groups, Algebra and Logic 7, 268–272 (1968).CrossRefGoogle Scholar
  91. [91]
    V. N. Remmeslennikov, Conjugacy in polycyclic groups, Algebra i Logica 8, 712725 (1969).Google Scholar
  92. [92]
    J. J. Rotman, “An introduction to the theory of groups”, (third edition), Allyn and Bacon,Boston, 1984.Google Scholar
  93. [93]
    I. N. Sanov, A property of a certain representation of a free group, Dokl. Akad. Nauk SSSR 57, 657–659 (1947).MATHMathSciNetGoogle Scholar
  94. [94]
    E. A. Scott, A finitely presented simple group with unsolvable conjugacy problem, Journ. Algebra 90, 333–353 (1984).CrossRefMathSciNetGoogle Scholar
  95. [95]
    E. A. Scott, A tour around finitely presented simple groups,this volume.Google Scholar
  96. [96]
    P. E. Schupp, On Dehn’s algorithm and the conjugacy problem, Math. Ann. 178, 119–130 (1968).CrossRefMATHMathSciNetGoogle Scholar
  97. [97]
    P. E. Schupp, Embeddings into simple groups, J. London Math. Soc. 13, 90–94 (1976).CrossRefMATHMathSciNetGoogle Scholar
  98. [98]
    D. Segal, “Polycyclic groups ”, Cambridge University Press, 1983Google Scholar
  99. [99]
    D. Segal, Decidable properties of polycyclic groups,to appear.Google Scholar
  100. [100]
    C. C. Squier, Word problems and a homological finiteness condition for monoids, J. Pure Appl. Algebra 49, 201–217 (1989).Google Scholar
  101. [101]
    V. A. Tartakovskii, The sieve method in group theory, Mat. Sbornik 25, 350 (1949).Google Scholar
  102. [102]
    V. A. Tartakovskii, Application of the sieve method to the solution of the word problem for certain types of groups, Mat. Sbornik 25, 251–274 (1949).Google Scholar
  103. [103]
    V. A. Tartakovskii, Solution of the word problem for groups with a k-reduced basis for k6, Izv. Akad. Nauk SSSR Ser. Math. 13, 483–494 (1949).MATHGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1992

Authors and Affiliations

  • Charles F. MillerIII
    • 1
  1. 1.Department of MathematicsUniversity of MelbourneParkvilleAustralia

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