Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

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

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


The word problem and the isomorphism problem for groups
HTML articles powered by AMS MathViewer

by John Stillwell PDF
Bull. Amer. Math. Soc. 6 (1982), 33-56
  • 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 10.2307/1970103
  • J. L. Britton, The word problem for groups, Proc. London Math. Soc. (3) 8 (1958), 493–506. MR 125019, DOI 10.1112/plms/s3-8.4.493
  • John L. Britton, The word problem, Ann. of Math. (2) 77 (1963), 16–32. MR 168633, DOI 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), Studies in Logic and the Foundations of Mathematics, 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) Lecture Notes in Math., Vol. 450, Springer, Berlin, 1975, pp. 1–62. 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 10.1007/BF01455155
  • M. Dehn, Die beiden Kleeblattschlingen, Math. Ann. 75 (1914), no. 3, 402–413 (German). MR 1511799, DOI 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 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 10.1007/BF02395059
  • Hans Hermes, Aufzählbarkeit, Entscheidbarkeit Berechenbarkeit: Einführung in die Theorie der rekursiven Funktionen, Die Grundlehren der mathematischen Wissenschaften, Band 109, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1961 (German). MR 0143699, DOI 10.1007/978-3-662-01462-2
  • G. Higman, Subgroups of finitely presented groups, Proc. Roy. Soc. London Ser. A 262 (1961), 455–475. MR 130286, DOI 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 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 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 10.2307/1970290
  • Marvin L. Minsky, Computation: finite and infinite machines, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. 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 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, Izdat. Akad. Nauk SSSR, Moscow, 1955 (Russian). Trudy Mat. Inst. Steklov. no. 44. 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 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 10.2307/2267170
  • Michael O. Rabin, Recursive unsolvability of group theoretic problems, Ann. of Math. (2) 67 (1958), 172–194. MR 110743, DOI 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) Lecture Notes in Math., Vol. 372, Springer, Berlin, 1974, pp. 611–632. 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, DOI 10.1007/978-1-4684-0110-3
  • Heinrich Tietze, Über die topologischen Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatsh. Math. Phys. 19 (1908), no. 1, 1–118 (German). MR 1547755, DOI 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 10.2307/1969481
  • B. L. van der Waerden, Free products of groups, Amer. J. Math. 70 (1948), 527–528. MR 25465, DOI 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 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
Additional Information
  • Journal: Bull. Amer. Math. Soc. 6 (1982), 33-56
  • MSC (1980): Primary 01A60, 20F05, 20F10; Secondary 03D10
  • DOI:
  • MathSciNet review: 634433