Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

   
 
 

 

The word problem and the isomorphism problem for groups


Author: John Stillwell
Journal: Bull. Amer. Math. Soc. 6 (1982), 33-56
MSC (1980): Primary 01A60, 20F05, 20F10; Secondary 03D10
DOI: https://doi.org/10.1090/S0273-0979-1982-14963-1
MathSciNet review: 634433
Full-text PDF Free Access

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • S. I. Adyan, Unsolvability of some algorithmic problems in the theory of groups., Trudy Moskov. Mat. Obšč. 6 (1957), 231–298 (Russian). MR 0095872
  • Artin, E., 1926: Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg 4, 47-72. Boone, W. W., 1954 a, b; 1955 a, b: Certain simple, unsolvable problems in the theory of groups. I, II, III, IV, Nederl. Akad. Wetensch. Proc. Ser. A. 57, 231-237, 492-497; 58, 252-256, 571-577.
  • William W. Boone, The word problem, Ann. of Math. (2) 70 (1959), 207–265. MR 179237, DOI https://doi.org/10.2307/1970103
  • J. L. Britton, The word problem for groups, Proc. London Math. Soc. (3) 8 (1958), 493–506. MR 125019, DOI https://doi.org/10.1112/plms/s3-8.4.493
  • John L. Britton, The word problem, Ann. of Math. (2) 77 (1963), 16–32. MR 168633, DOI https://doi.org/10.2307/1970200
  • Stȧl Aanderaa and Daniel E. Cohen, Modular machines, the word problem for finitely presented groups and Collins’ theorem, Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), Stud. Logic Foundations Math., vol. 95, North-Holland, Amsterdam-New York, 1980, pp. 1–16. MR 579934
  • Reminiscences of logicians, Algebra and logic (Fourteenth Summer Res. Inst., Austral. Math. Soc., Monash Univ., Clayton, 1974) Springer, Berlin, 1975, pp. 1–62. Lecture Notes in Math., Vol. 450. Reported by J. N. Crossley; With contributions by C. C. Chang, John Crossley, Jerry Keisler, Steve Kleene, Mike Morley, Vivienne Morley, Andrzej Mostowski, Anil Nerode, Gerald Sacks, Peter Hilton and David Lucy. MR 0379111
  • Martin Davis (ed.), The undecidable. Basic papers on undecidable propositions, unsolvable problems and computable functions, Raven Press, Hewlett, N.Y., 1965. MR 0189996
  • M. Dehn, Über die Topologie des dreidimensionalen Raumes, Math. Ann. 69 (1910), no. 1, 137–168 (German). MR 1511580, DOI https://doi.org/10.1007/BF01455155
  • M. Dehn, Die beiden Kleeblattschlingen, Math. Ann. 75 (1914), no. 3, 402–413 (German). MR 1511799, DOI https://doi.org/10.1007/BF01563732
  • Kurt Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatsh. Math. Phys. 38 (1931), no. 1, 173–198 (German). MR 1549910, DOI https://doi.org/10.1007/BF01700692
  • Geoffrey Hemion, On the classification of homeomorphisms of $2$-manifolds and the classification of $3$-manifolds, Acta Math. 142 (1979), no. 1-2, 123–155. MR 512214, DOI https://doi.org/10.1007/BF02395059
  • Hans Hermes, Aufzählbarkeit, Entscheidbarkeit Berechenbarkeit: Einführung in die Theorie der rekursiven Funktionen, Die Grundlehren der mathematischen Wissenschaften, Bd. 109, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961 (German). MR 0143699
  • G. Higman, Subgroups of finitely presented groups, Proc. Roy. Soc. London Ser. A 262 (1961), 455–475. MR 130286, DOI https://doi.org/10.1098/rspa.1961.0132
  • Graham Higman, B. H. Neumann, and Hanna Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247–254. MR 32641, DOI https://doi.org/10.1112/jlms/s1-24.4.247
  • Stephen C. Kleene, Origins of recursive function theory, 20th Annual Symposium on Foundations of Computer Science (San Juan, Puerto Rico, 1979) IEEE, New York, 1979, pp. 371–382. MR 598119
  • Klein, F., 1882: Letter to Poincare, 14 May 1882, Ges. Math. Abh. III, 615-616. Magnus, W., 1930: Über diskontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz), J. Reine Angew. Math. 163, 141-165.
  • W. Magnus, Das Identitätsproblem für Gruppen mit einer definierenden Relation, Math. Ann. 106 (1932), no. 1, 295–307 (German). MR 1512760, DOI https://doi.org/10.1007/BF01455888
  • Magnus, W., Karrass, A. and Solitar, D., 1966: Combinatorial group theory, Interscience, New York.
  • B. Chandler and V. Magnus, Razvitie kombinatornoĭ teorii grupp, “Mir”, Moscow, 1985 (Russian). Ocherk istorii razvitiya ideĭ. [A case study in the history of ideas]; Translated from the English by A. L. Semenov; Translation edited and with a preface by S. I. Adyan. MR 835915
  • A. Markoff, On the impossibility of certain algorithms in the theory of associative systems, C. R. (Doklady) Acad. Sci. URSS (N.S.) 55 (1947), 583–586. MR 0020528
  • A. A. Markov, Insolubility of the problem of homeomorphy, Proc. Internat. Congress Math. 1958, Cambridge Univ. Press, New York, 1960, pp. 300–306 (Russian). MR 0115160
  • Marvin L. Minsky, Recursive unsolvability of Post’s problem of “tag” and other topics in theory of Turing machines, Ann. of Math. (2) 74 (1961), 437–455. MR 140405, DOI https://doi.org/10.2307/1970290
  • Marvin L. Minsky, Computation: finite and infinite machines, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. Prentice-Hall Series in Automatic Computation. MR 0356580
  • B. H. Neumann, An essay on free products of groups with amalgamations, Philos. Trans. Roy. Soc. London Ser. A 246 (1954), 503–554. MR 62741, DOI https://doi.org/10.1098/rsta.1954.0007
  • Nielsen, J., 1921: Om Regning med ikke kommutative Faktoren og dens Anvendelse i Gruppeteorien, Mat. Tidsskr. B, 77-94.
  • P. S. Novikov, Ob algoritmičeskoĭ nerazrešimosti problemy toždestva slov v teorii grupp, Trudy Mat. Inst. Steklov. no. 44, Izdat. Akad. Nauk SSSR, Moscow, 1955 (Russian). MR 0075197
  • P. S. Novikov, Über einige algorithmische Probleme der Gruppentheorie, Jber. Deutsch. Math.-Verein. 61 (1958), no. Abt. 1, 88–92 (German). MR 107664
  • Poincaré, H., 1892: Sur l’Analysis situs. Comptes Rendus 115, 633-636. Poincaré, H., 1895: Analysis situs, J. de l’École Polytech. (2) 1, 1-123. Post, E., 1936: Finite combinatory processes—formulation. I, J. Symbolic Logic 1, 103-105.
  • Emil L. Post, A variant of a recursively unsolvable problem, Bull. Amer. Math. Soc. 52 (1946), 264–268. MR 15343, DOI https://doi.org/10.1090/S0002-9904-1946-08555-9
  • Emil L. Post, Recursive unsolvability of a problem of Thue, J. Symbolic Logic 12 (1947), 1–11. MR 20527, DOI https://doi.org/10.2307/2267170
  • Michael O. Rabin, Recursive unsolvability of group theoretic problems, Ann. of Math. (2) 67 (1958), 172–194. MR 110743, DOI https://doi.org/10.2307/1969933
  • Schreier, O., 1927: Die Untergruppen der freien Gruppen, Abh. Math. Sem. Univ. Hamburg 5, 161-183.
  • Paul E. Schupp, Some reflections on HNN extensions, Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973) Springer, Berlin, 1974, pp. 611–632. Lecture Notes in Math., Vol. 372. MR 0357617
  • John C. Stillwell, Classical topology and combinatorial group theory, Graduate Texts in Mathematics, vol. 72, Springer-Verlag, New York-Berlin, 1980. MR 602149
  • Heinrich Tietze, Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatsh. Math. Phys. 19 (1908), no. 1, 1–118 (German). MR 1547755, DOI https://doi.org/10.1007/BF01736688
  • Turing, A. M., 1936: On computable numbers, with an application to the Entscheidungsproblem, Proc. London Math. Soc. (2) 42, 230-265.
  • A. M. Turing, The word problem in semi-groups with cancellation, Ann. of Math. (2) 52 (1950), 491–505. MR 37294, DOI https://doi.org/10.2307/1969481
  • B. L. van der Waerden, Free products of groups, Amer. J. Math. 70 (1948), 527–528. MR 25465, DOI https://doi.org/10.2307/2372196
  • Friedhelm Waldhausen, The word problem in fundamental groups of sufficiently large irreducible $3$-manifolds, Ann. of Math. (2) 88 (1968), 272–280. MR 240822, DOI https://doi.org/10.2307/1970574
  • Friedhelm Waldhausen, Recent results on sufficiently large $3$-manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 21–38. MR 520520

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1980): 01A60, 20F05, 20F10, 03D10

Retrieve articles in all journals with MSC (1980): 01A60, 20F05, 20F10, 03D10


Additional Information