Bulletin (New Series) of the American Mathematical Society

The word problem and the isomorphism problem for groups

John Stillwell

Full-text: Open access

Article information

Source
Bull. Amer. Math. Soc. (N.S.), Volume 6, Number 1 (1982), 33-56.

Dates
First available in Project Euclid: 4 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.bams/1183548590

Mathematical Reviews number (MathSciNet)
MR634433

Zentralblatt MATH identifier
0483.20018

Subjects
Primary: 01A60: 20th century 20F05: Generators, relations, and presentations 20F10: Word problems, other decision problems, connections with logic and automata [See also 03B25, 03D05, 03D40, 06B25, 08A50, 20M05, 68Q70]
Secondary: 03D10: Turing machines and related notions [See also 68Q05]

Citation

Stillwell, John. The word problem and the isomorphism problem for groups. Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 1, 33--56. https://projecteuclid.org/euclid.bams/1183548590


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References

  • Adian, S. I., 1957: The unsolvability of certain algorithmic problems in the theory of groups Trudy Moskov. Mat. Obsc. 6, 231-298. (Russian)
  • 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.
  • Boone, W. W., 1959: The word problem, Ann. of Math. (2) 70, 207-265.
  • Britton, J. L., 1958: The word problem for groups, Proc. London Math. Soc. 8, 493-506.
  • Britton, J. L., 1963: The word problem, Ann. of Math. (2) 77, 16-32.
  • Cohen, D. E., and Aanderaa, S., 1980: Modular machines, the word problem for finitely presented groups, and Collins' theorem, Word Problems II (S. I. Adian, W. W. Boone, G. Higman, eds.), North-Holland, Amsterdam, pp. 1-16.
  • Crossley, J. N., 1975: Reminiscences of logicians, Algebra and Logic (J. N. Crossley, ed.), Lecture Notes in Math., vol. 450, Springer-Verlag, Berlin and New York, 1-62.
  • Davis, M., 1965: The undecidable, Raven Press, New York.
  • Dehn, M., 1910: Über die Topologie des dreidimensional Raumes, Math. Ann. 69, 137-168.
  • Dehn, M., 1914: Die beiden Kleeblattschlingen, Math. Ann. 75, 402-413.
  • Gödel, K., 1931: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I, Monatsh. f. Math. u. Phys. 38, 173-198.
  • Hemion, G., 1979: On the classification of homeomorphisms of 2-manifolds and the classification of 3-manifolds, Acta Math. 142, 123-155.
  • Hermes, H., 1961: Aufzählbarkeit, Entscheidbarkeit, Berechenbarkeit, Springer-Verlag, Berlin and New York.
  • Higman, G., 1961: Subgroups of finitely presented groups, Proc. Roy. Soc. Ser. A. 262, 455-475.
  • Higman, G., Neumann, B. H. and Neumann, H., 1949: Embedding theorems for groups, J. London Math. Soc. 24, 247-254.
  • Kleene, S. C., 1979: Origins of recursive function theory, IEEE 20th Annual Symposium on Foundations of Computer Science, pp. 371-382.
  • 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.
  • Klein, F., 1932: Das Identitätsproblem für Gruppen mit einer definierenden Relation, Math. Ann. 106, 295-307.
  • Magnus, W., Karrass, A. and Solitar, D., 1966: Combinatorial group theory, Interscience, New York.
  • Magnus, W. and Chandler, B., History of combinatorial group theory (to appear).
  • Markov, A. A., 1947: On the impossibility of certain algorithms in the theory of associative systems, Dokl. Akad. Sci. USSR 55, 583-586. (Russian)
  • Markov, A. A., 1958: Insolubility of the problem of homeomorphy, Proc. Internat. Congr. Math., pp. 300-306. (Russian)
  • Minsky, M., 1961: Recursive unsolvability of Post's problem of 'Tag' and other topics in the theory of Turing machines, Ann. of Math. (2) 74, 437-445.
  • Minsky, M., 1967: Computation: finite and infinite machines, Prentice-Hall, Englewood Cliffs, N. J.
  • Neumann, B. H., 1954: An essay on free products of groups with amalgamation, Philos. Trans. Roy. Soc. London Ser. A. 246, 503-554.
  • Nielsen, J., 1921: Om Regning med ikke kommutative Faktoren og dens Anvendelse i Gruppeteorien, Mat. Tidsskr. B, 77-94.
  • Novikov, P. S., 1955: On the algorithmic unsolvability of the word problem in group theory, Trudy Mat. Inst. Steklov 44, 143 pp. (Russian)
  • Novikov, P. S., 1958: Über einige algorithmische Probleme der Gruppentheorie, Jber. Deutsch. Math. Verein 61, 88-92.
  • 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.
  • Post, E., 1946: A variant of a recursively unsolvable problem, Bull. Amer. Math. Soc. 52, 264-268.
  • Post, E., 1947: Recursive unsolvability of a problem of Thue, J. Symbolic Logic 12, 1-11.
  • Rabin, M. O., 1958: Recursive unsolvability of group theoretic problems, Ann. of Math. (2) 67, 172-194.
  • Schreier, O., 1927: Die Untergruppen der freien Gruppen, Abh. Math. Sem. Univ. Hamburg 5, 161-183.
  • Schupp, P. E., 1974: Some reflections on HNN extensions, Proc. 2nd Internat. Conf. Theory of Groups (M. F. Newman, ed.), Lecture Notes in Math., vol. 372, Springer-Verlag, Berlin and New York, pp. 611-632.
  • Stillwell, J. C., 1980: Classical topology and combinatorial group theory, Springer-Verlag, Berlin and New York.
  • Tietze, H., 1908: Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatsh. f. Math. u. Phys. 19, 1-118.
  • Turing, A. M., 1936: On computable numbers, with an application to the Entscheidungsproblem, Proc. London Math. Soc. (2) 42, 230-265.
  • Turing, A. M., 1950: The word problem in semigroups with cancellation, Ann. of Math. (2) 52, 491-505.
  • van der Waerden, B. L., 1948: Free products of groups, Amer. J. Math. 70, 527-528.
  • Waldhausen, F., 1968: The word problem in fundamental groups of sufficiently large irreducible 3-manifolds, Ann. of Math. (2) 88, 272-280.
  • Waldhausen, F., 1978: Recent results on sufficiently large 3-manifolds, Proc. Sympos. Pure Math., vol. 32, Part 2, Amer. Math. Soc., Providence, R. I., pp. 21-38.