Abstract
The following result is proved. Let n be a positive integer and G a residually finite group in which every product of at most 68 commutators has order dividing n. Then G′ is locally finite.
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S. I. Adian, The Burnside problem and identities in groups (Translated from the Russian by John Lennox and JamesWiegold), Ergebnisse der Mathematik und ihrer Grenzgebiete, 95, Springer-Verlag, Berlin-New York, 1979.
S. I. Adian, On groups with periodic commutators, Doklady Mathematics, 62 (2000), 174–176.
S. V. Aleshin, Finite automata and the Burnside problem for periodic groups, Mathematical Notes 11 (1972), 199–203.
R. Brandl, Commutators and π-subgroups, Proceedings of the American Mathematical Society 109 (1990), 305–308.
G. S. Deryabina and P. A. Kozhevnikov, The derived subgroup of a group with commutators of bounded order can be non-periodic, Commuications in Algebra 27(9) (1999), 4525–4530.
W. Feit and J. Thompson, Solvability of groups of odd order, Pacific Journal of Mathematics 13 (1963), 773–1029.
E. S. Golod, On nil-algebras and residually finite groups, Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya 28 (1964), 273–276.
R. I. Grigorchuk, On the Burnside problem for periodic groups, Functional Analysis and its Applications 14 (1980), 53–54.
N. D. Gupta, Periodicity of the commutator subgroup of a certain group, Notices of the American Mathematical Society 14 (1967), 703.
N. Gupta and S. Sidki, On the Burnside problem for periodic groups, Mathematische Zeitschrift 182 (1983), 385–386.
M. Hall, The Theory of Groups, Macmillan, New York, 1959.
P. Hall and G. Higman, The p-length of a p-soluble group and reduction theorems for Burnside’s problem, Proceedings of the London Mathematical Society (3) 6 (1956), 1–42.
G. A. Jones, Varieties and simple groups, Journal of the Australian Mathematical Society 17 (1974), 163–173.
E. I. Khukhro and V. D. Mazurov, eds., Kourovka Notebook, 13th Edition, Novosibirsk, 1994.
I. D. MacDonald, On certain varieties of groups, Mathematische Zeitschrift 76 (1961), 270–282.
N. S. Mendelsohn, Some examples of man-machine interaction in the solution of mathematical problems, in Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967), Pergamon, Oxford, 1970, pp. 217–222.
N. Nikolov and D. Segal, On finitely generated profinite groups, I: strong completeness and uniform bounds, Annals of Mathematics 165 (2007), 171–238.
A. Yu. Olshanskii, Geometry of Defining Relations in Groups, Translated from the 1989 Russian original by Yu. A. Bakhturin, Mathematics and its Applications (Soviet Series), 70, Kluwer Academic Publishers Group, Dordrecht, 1991.
D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Part 1, Springer-Verlag, Berlin-New York, 1972.
D. Segal, Closed subgroups of profinite groups, Proceedings of the London Mathematical Society (3) 81 (2000), 29–54.
P. Shumyatsky, Groups with commutators of bounded order, Proceedings of the American Mathematical Society 127 (1999), 2583–2586.
P. Shumyatsky, Commutators in residually finite groups, Monatshefte für Mathematik 137 (2002), 157–165.
P. Shumyatsky, On varieties arising from the solution of the Restricted Burnside Problem, Journal of Pure and Applied Algebra 171 (2002), 67–74.
P. Shumyatsky, On the Fitting height of a finite group, Journal of Group Theory 13 (2010), 139–142.
P. Shumyatsky and J. C. Silva, The Restricted Burnside Problem for multilinear commutators, Mathematical Proceedings of the Cambridge Philosophical Society 146 (2009), 603–613.
P. Shumyatsky and J. C. Silva, Engel Words and the Restricted Burnside Problem, Monatshfte für Mathematik 159 (2010), 397–405.
V.I. Sushchansky, Periodic p-elements of permutations and the general Burnside problem, Doklady Akademii Nauk SSSR 247 (1979), 447–461.
E. Zelmanov, The solution of the restricted Burnside problem for groups of odd exponent, Mathematics of USSR-Izvestiya 36 (1991), 41–60.
E. Zelmanov, The solution of the restricted Burnside problem for 2-groups, Mathematics of the USSR-Sbornik 182 (1991), 568–592.
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Shumyatsky, P. Commutators in residually finite groups. Isr. J. Math. 182, 149–156 (2011). https://doi.org/10.1007/s11856-011-0027-3
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DOI: https://doi.org/10.1007/s11856-011-0027-3