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Multilinear commutators in residually finite groups

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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.

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References

  1. S. I. Adian, On groups with periodic commutators, Doklady Mathematics 62 (2000), 174–176.

    Google Scholar 

  2. S. V. Aleshin, Finite automata and the Burnside problem for periodic groups, Mathematical Notes 11 (1972), 199–203.

    Article  MATH  Google Scholar 

  3. 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.

    Article  MathSciNet  MATH  Google Scholar 

  4. 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.

    Article  MathSciNet  MATH  Google Scholar 

  5. W. Feit and J. Thompson, Solvability of groups of odd order, Pacific Journal of Mathematics 13 (1963), 773–1029.

    Google Scholar 

  6. P. Flavell, S. Guest and R. Guralnick, Characterizations of the solvable radical, Proceedings of the American Mathematical Society 138 (2010), 1161–1170.

    Article  MathSciNet  MATH  Google Scholar 

  7. E. S. Golod, On nil-algebras and residually finite groups, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 28 (1964), 273–276.

    MathSciNet  Google Scholar 

  8. 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.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. I. Grigorchuk, On the Burnside problem for periodic groups, Functional Analysis and its Applications 14 (1980), 53–54.

    MathSciNet  Google Scholar 

  10. N. Gupta and S. Sidki, On the Burnside problem for periodic groups, Mathematische Zeitschrift 182 (1983), 385–386.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Hall, The Theory of Groups, Macmillan, New York, 1959.

    MATH  Google Scholar 

  12. 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.

    Article  MathSciNet  MATH  Google Scholar 

  13. B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, 1967.

    Book  MATH  Google Scholar 

  14. G. A. Jones, Varieties and simple groups, Journal of the Australian Mathematical Society 17 (1974), 163–173.

    Article  MATH  Google Scholar 

  15. N. Nikolov and D. Segal, On finitely generated profinite groups, I: strong completeness and uniform bounds, Annals of Mathematics 165 (2007), 171–238.

    Article  MathSciNet  MATH  Google Scholar 

  16. D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Part 1, Springer-Verlag, Berlin-New York, 1972.

    Google Scholar 

  17. D. Segal, Closed subgroups of profinite groups, Proceedings of the London Mathematical Society. Third Series 81 (2000), 29–54.

    Article  MathSciNet  MATH  Google Scholar 

  18. P. Shumyatsky, Groups with commutators of bounded order, Proceedings of the American Mathematical Society 127 (1999), 2583–2586.

    Article  MathSciNet  MATH  Google Scholar 

  19. P. Shumyatsky, Verbal subgroups in residually finite groups, The Quarterly Journal of Mathematics 51 (2000), 523–528; doi:10.1093/qjmath/51.4.523.

    Article  MathSciNet  MATH  Google Scholar 

  20. P. Shumyatsky, On varieties arising from the solution of the Restricted Burnside Problem, Journal of Pure and Applied Algebra 171 (2002), 67–74.

    Article  MathSciNet  MATH  Google Scholar 

  21. P. Shumyatsky, Commutators in residually finite groups, Monatshefte für Mathematik 137 (2002), 157–165.

    Article  MathSciNet  MATH  Google Scholar 

  22. P. Shumyatsky and J. C. Silva, The Restricted Burnside Problem for multilinear commutators, Mathematical Proceedings of the Cambridge Philosophical Society 146 (2009), 603–613.

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Shumyatsky, Commutators in residually finite groups, Israel Journal of Mathematics 182 (2011), 149–156.

    Article  MathSciNet  MATH  Google Scholar 

  24. V. I. Sushchansky, Periodic p-elements of permutations and the general Burnside problem, Rossiĭskaya Akademiya Nauk 247 (1979), 447–461.

    Google Scholar 

  25. E. Zelmanov, The solution of the restricted Burnside problem for groups of odd exponent, Math. USSR Izv. 36 (1991), 41–60.

    Article  MathSciNet  Google Scholar 

  26. E. Zelmanov, The solution of the restricted Burnside problem for 2-groups, Sbornik Mathematics 182 (1991), 568–592.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900, Brazil

    Pavel Shumyatsky

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  1. Pavel Shumyatsky
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Correspondence to Pavel Shumyatsky.

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Supported by CNPq-Brazil.

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Shumyatsky, P. Multilinear commutators in residually finite groups. Isr. J. Math. 189, 207–224 (2012). https://doi.org/10.1007/s11856-011-0157-7

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  • DOI: https://doi.org/10.1007/s11856-011-0157-7

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