References
Appel, K. I.,On two-variable equations in free groups, Proc. Amer. Math. Soc., 21 (1969) 179–181.
Baudisch, A.,Kommutatorsgleichungen in semifreiern Gruppen, Acta Math. Acad. Sci. Hungar. 29 (1977) 235–249.
Baumslag, G., Mahler, K.,Equations in free metabelian groups, Mich. Math. J. 12 (1963) 375–379.
Baumslag, G., Steinberg, A.,Residual nilpotence and relations in free groups, Bull. Amer. Math. Soc. 70 (1964) 283–284.
Brenner, J. L., Lim, M. J. S.,The matrix equations A=XYZ and B=ZYX and related ones, Canad. Math. Bull. 17 (1974) 179–183.
Bulitko, V. K.,Equations and inequalities in a free group and a free semigroup, Tul. Gos. Ped. Inst. Ucen. Zap Mat. Kaf.Vyp 2 Geom. Alg 1972, 242–252.
Burns, R. G., Edmunds, C. C., Formanek, E.,The equations s −1 t −1 st=u −1 v −1 uv in a free product of groups, Math. Z. 153 (1977) 83–88.
Burns, R. G., Edmunds, C. C., Farouqi, I. H.,On commutator equalities and stabilizers in free groups, Canad. Math. Bull. 19 (1976) 263–267.
Dark, R. S., Newell, M. L.,On conditions for commutators to form a subgroup, J. London Math. Soc. 17 (1978) 251–262.
Dey, I. M. S., Wiegold, J.,Generators for alternating and symmetric groups, Austral. J. Math. 12 (1971) 63–68.
Edmunds, C. C.,On the endomorphism problem for free groups, Comm. Algebra 3 (1975) 7–20.
Edmunds, C. C.,Products of commutators as products of squares, Canad. J. Math. 27 (1975) 1329–1338.
Edmunds, C. C.,A short combinatorial proof of the Vaught conjecture, Canad. Bull. Math. 18 (1975) 607–608.
Edmunds, C. C.,On the endomorphism problem for free groups II. Proc. London Math. Soc. 38 (1979) 153–168.
Edmunds, C. C.,On conditions equivalent to the solvability of the endomorphism problem for free groups. To appear.
Finkelstein, H., Mandelberg, K. I.,On solutions of equations in symmetric groups, J. Combinatorial theory A25 (1978) 142–152.
Flanders, H.,Elementary divisors of ABand BA, Proc. Amer. Math. Soc. 2 (1951) 871–874.
Gerstenhaber, M., Rothaus, O. S.,The solution of sets of equations in groups, Proc. Nat. Acad. Sci. U.S.A. 48 (1962) 1531–1533.
Guralnick, R. M.,On groups with decomposable commutator subgroups, Glasgow Math. J. 19 (1978) 159–162.
Guralnick, R. M.,On cyclic commutator groups. To appear.
Higman, G., Neumann, B. H., Neumann, H.,Embedding theorems for groups. J. London Math. Soc. 24 (1949) 247–254.
Itô, N.,A theorem on the alternating group A n (n≥5), Math. Japon. 2 (1951) 59–60.
Lorents, A. A.,Solution of systems of equations in one unknown in free groups, Dokl. Akad. Nauk SSSR 148 (1963) 1253–1256.
Lorents, A. A.,Coefficient-free equations in free groups, Dokl. Akad. Nauk SSSR 160 (1965) 538–540.
Lorents, A. A.,Representation of sets of solutions of systems of equations with one unknown in a free group, Dokl. Akad. Nauk SSSR 178 (1968) 290–292.
Lyndon, R. C.,Equations in free groups, Trans. Amer. Math. Soc. 96 (1960) 445–457.
Lyndon, R. C.,Groups with parametric exponents, Trans. Amer. Math. Soc., 96 (1960) 518–533.
Lyndon, R. C.,Equations in free, metabelian groups, Proc. Amer. Math. Soc. 17 (1966) 728–730.
Lyndon, R. C.,Quadratic equation in free groups with amalgamation, Houston J. Math.
Lyndon, R. C., McDonough, T., Newman, M.,On products of powers in groups, Proc. Amer. Math. Soc. 40 (1973) 419–420.
Lyndon, R. C., Schupp, P. E.,Combinatorial group theory, Ergebnisse der Math. 89 (1977).
Lyndon, R. C., Schützenberger, M. P.,The equation a M=b N c P in a free group, Mich. Math. J. 9 (1962) 289–298.
Lyndon, R. C. Wicks, M. J.,Commutators in free groups, To appear.
Macintyre, A.,On algebraically closed groups, Ann. of Math. 96 (1972) 53–97.
Magnus, W., Karrass, A., Solitar, D.,Combinatorial group theory, Wiley, New York, 1966.
Makanin, G. S.,The rank of coefficient-free equations with four unknowns in a free semigroup, Mat. Sb. 100 (1976) 285–311.
Makanin, G. S.,The problem of the solvability of equations in a free semigroup, Mat. Sb. 103 (1976) 147–236, 319.
McCool, J.,Some finitely presented subgroups of the automorphism group of a free group, J. Algebra 35 (1975) 265–270.
Miller, C. F. III.
Mycielski, J.,Can one solve equations in groups?, Amer. Math. Monthly 84 (1977) 723–726.
Neumann, B. H.,Adjunction of elements to groups, J. London Math. Soc. 18 (1943) 4–11.
Neumann, B. H.,An essay on free products of groups with amalgamations, Philos. Trans. Roy. Soc. London Ser. A 246 (1954) 303–554.
Neumann, B. H.,The isomorphism problem for algebraically closed groups, Word Problems, North-Holland, Amsterdam, 1973, 553–562.
Ore, O.,Some remarks on commutators, Proc. Amer. Math. Soc. 2 (1951) 307–314.
Qin, Jian-Min.,On commutators in orthogonal groups, Acta Math. Sinica, 15 (1969) 708–719.
Ree, R.,Commutators in semisimple algebraic groups, Proc. Amer. Math. Soc. 15 (1964) 457–460.
Rodney, D. M.,Commutators and abelian groups, J. Austral. Math. Soc. 24 (1977) 79–91.
Rosenberger, G.,Produkte von Potenzen und Kommutatoren in freien Gruppen, J. Alg.
Rosenberger, G.Alternierende Produkte in freien Gruppen. I11.J. Math.
Rothaus, O. S.,On the non-triviality of some group extensions given by generators and relations, Ann. of Math. 106 (1977) 599–612.
Schupp, P. E.,On the substitution problen for free groups, Proc Amer. Math. Soc. 23 (1969) 421–423.
Scott, W. R.,Algebraically closed groups, Proc. Amer. Math. Soc. 2 (1951) 118–121.
Stallings, J. R.
Steinberg, A.,On equations in free groups, Mich. Math. J. 18 (1971) 87–95.
Taussky-Todd, O.,Problems on matrices and operators, Bull. Amer. Math. Soc. 76 (1970) 977.
Thompson, R. C.,Commutators in the special and general linear groups, Trans. Amer. Math. Soc. 101 (1961) 16–33.
Thompson, R. C.,Commutators of matrices with coefficients from the field of two elements, Duke Math. J. 29 (1962) 367–374.
Thompson, R. C.,On matrix commutators, Port. Math. 21 (1962) 143–153.
Warlimont, R.,Über die Anzahl der Lösungen von x n=1 in der symmetrischer Gruppe S n , Archiv der Math. 30 (1978) 591–594.
Wicks, M. J.,Commutators in free products, J. London Math. Soc. 37 (1962) 433–444.
Wicks, M. J.,A relation in free products, Proc. Cong. Canberra 1973, Springer Lecture Notes 372 (1974) 709–716.
Zieschang, H.,Über Worte S 1 a 1 S 2 a 2 ...S p a p in einer freien Gruppen mit p freien Erzeugenden, Math. Ann. 147 (1962) 143–153.
Brenner, J. L., Some simultaneous equations in matrices, Duke Math J. 39 (1972) 801–805.
Culler, M., Using surfaces to solve equations in free groups, Topology. To appear.
Gallagher, P. X., Group characters and commutators, Math. Z. 79 (1962) 122–126.
Guralnick, R. M., On a result of Schur, J. Algebra 59 (1979) 302–310.
Rosenlicht, M., On a result of Baer, Proc. Amer. Math. Soc. 13 (1962) 99–101.
Author information
Authors and Affiliations
Additional information
This work was supported in part by the National Science Foundation, and partly by CNPq and FINEP while the author was visiting at the Universidade de Brasilia.
About this article
Cite this article
Lyndon, R.C. Equations in groups. Bol. Soc. Bras. Mat 11, 79–102 (1980). https://doi.org/10.1007/BF02584882
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02584882