Abstract
The following result is proved. Let w be a multilinear commutator and n a positive integer. Suppose that G is a residually finite group in which every product of at most 896 w-values has order dividing n. Then the verbal subgroup w(G) is locally finite.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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.
A. Al-Roqi and P. Flavell, On the Fitting height of a soluble group that is generated by a conjugacy class of 3-elements, The Bulletin of the London Mathematical Society 39 (2007), 973–981.
G. S. Deryabina and P. A. Kozhevnikov, The derived subgroup of a group with commutators of bounded order can be non-periodic, Communications in Algebra 27 (1999), 4525–4530.
W. Feit and J. Thompson, Solvability of groups of odd order, Pacific Journal of Mathematics 13 (1963), 773–1029.
P. Flavell, S. Guest and R. Guralnick, Characterizations of the solvable radical, Proceedings of the American Mathematical Society 138 (2010), 1161–1170.
E. S. Golod, On nil-algebras and residually finite groups, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 28 (1964), 273–276.
N. Gordeev, F. Grunewald, B. Kunyavskii and E. Plotkin, A description of Baer-Suzuki type of solvable radical of a finite group, Journal of Pure and Applied Algebra 213 (2009), 250–258.
R. I. Grigorchuk, On the Burnside problem for periodic groups, Functional Analysis and its Applications 14 (1980), 53–54.
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. Third Series 6 (1956), 1–42.
B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967.
G. A. Jones, Varieties and simple groups, Journal of the Australian Mathematical Society 17 (1974), 163–173.
N. Nikolov and D. Segal, On finitely generated profinite groups, I: strong completeness and uniform bounds, Annals of Mathematics 165 (2007), 171–238.
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. Third Series 81 (2000), 29–54.
P. Shumyatsky, Groups with commutators of bounded order, Proceedings of the American Mathematical Society 127 (1999), 2583–2586.
P. Shumyatsky, Verbal subgroups in residually finite groups, The Quarterly Journal of Mathematics 51 (2000), 523–528; doi:10.1093/qjmath/51.4.523.
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, Commutators in residually finite groups, Monatshefte für Mathematik 137 (2002), 157–165.
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, Commutators in residually finite groups, Israel Journal of Mathematics 182 (2011), 149–156.
V. I. Sushchansky, Periodic p-elements of permutations and the general Burnside problem, Rossiĭskaya Akademiya Nauk 247 (1979), 447–461.
E. Zelmanov, The solution of the restricted Burnside problem for groups of odd exponent, Math. USSR Izv. 36 (1991), 41–60.
E. Zelmanov, The solution of the restricted Burnside problem for 2-groups, Sbornik Mathematics 182 (1991), 568–592.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by CNPq-Brazil.
Rights and permissions
About this article
Cite this article
Shumyatsky, P. Multilinear commutators in residually finite groups. Isr. J. Math. 189, 207–224 (2012). https://doi.org/10.1007/s11856-011-0157-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-011-0157-7