Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-28T21:48:53.828Z Has data issue: false hasContentIssue false
  • English
  • Français

The restricted Burnside problem for multilinear commutators

Published online by Cambridge University Press:  01 May 2009

PAVEL SHUMYATSKY  and
JHONE CALDEIRA SILVA
PAVEL SHUMYATSKY
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900Brazil.
JHONE CALDEIRA SILVA
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia-DF, 70910-900Brazil.
Get access Rights & Permissions [Opens in a new window]

Abstract

The following theorem is proved. For any positive integer n there exists t depending only on n such that if the word w is a multilinear commutator, then the class of all groups G in which w(G) is locally finite and the product of any t values of w has order dividing n is a variety.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 3 , May 2009 , pp. 603 - 613
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Gorenstein, D.Finite Groups (Harper and Row, 1968).Google Scholar
[2]Hall, M.. The Theory of Groups (Macmillan, 1959).Google Scholar
[3]Hall, P. and Higman, G.. The p-length of a p-soluble group and reduction theorems for Burnside's problem. Proc. London Math. Soc. (3), 6 (1956), 1–42.CrossRefGoogle Scholar
[4]Huppert, B.. Endliche Gruppen I (Springer-Verlag, 1967).CrossRefGoogle Scholar
[5]Huppert, B. and Blackburn, N.. Finite Groups II (Springer-Verlag, 1982).CrossRefGoogle Scholar
[6]Jones, G. A.. Varieties and simple groups. J. Austral. Math. Soc. 17 (1974), 163–173.Google Scholar
[7]Khukhro, E. I. and Shumyatsky, P.. Bounding the exponent of a finite group with automorphisms. J. Algebra 212 (1999), 363–374.CrossRefGoogle Scholar
[8]Lazard, M.. Sur les groupes nilpotents et les anneaux de Lie. Ann. Sci. École Norm. Sup. 71 (1954), 101–190.Google Scholar
[9]Nikolov, N. and Segal, D.. On finitely generated profinite groups. I: strong completeness and uniform bounds. Ann. of Math. (2), 165 (2007), 171–238.Google Scholar
[10]Robinson, D. J. S.. Finiteness Conditions and Generalized Soluble Groups, Part 1 (Springer-Verlag, 1972).CrossRefGoogle Scholar
[11]Segal, D.. Closed subgroups of profinite groups. Proc. London Math. Soc. (3), 81 (2000), 29–54.Google Scholar
[12]Shumyatsky, P.. Applications of Lie ring methods to group theory, in: Nonassociative Algebra and its applications (Costa, R., Grishkov, A., Guzzo, H. Jr. and Peresi, L. A., eds). Marcel Dekker (2000), 373–395.Google Scholar
[13]Shumyatsky, P.. Verbal subgroups of residually finite groups. Quart. J. Math. (Oxford) 51 (2000), 523–528.Google Scholar
[14]Shumyatsky, P.. A (locally nilpotent)-by-nilpotent variety of groups. Math. Proc. Camb. Phil. Soc. 132 (2002), 193–196.Google Scholar
[15]Shumyatsky, P.. On varieties arising from the solution of the restricted Burnside problem. J. Pure Appl. Algebra 171 (2002), 67–74.Google Scholar
[16]Shumyatsky, P.. Commutators in residually finite groups. Monatsh. Math. 137 (2002), 157–165.Google Scholar
[17]Shumyatsky, P.. A variety of groups. Math. Proc. Camb. Phil. Soc. 138 (2005), 21–26.Google Scholar
[18]Wilson, J. S. and Zelmanov, E.. Identities for Lie algebras of pro-p groups. J. Pure. Appl. Algebra 81 (1992), 103–109.Google Scholar
[19]Zelmanov, E.. The solution of the Restricted Burnside Problem for groups of odd exponent. Math. USSR Izv. 36 (1991), 41–60.Google Scholar
[20]Zelmanov, E.. The solution of the Restricted Burnside Problem for 2-groups. Math. Sb. 182 (1991), 568–592.Google Scholar
[21]Zelmanov, E.. Nil rings and periodic groups. The Korean Math. Soc. Lecture Notes in Math. Seoul (1992).Google Scholar
[22]Zelmanov, E.. Lie ring methods in the theory of nilpotent groups. In: Campbell, C. M. et al. (Eds.), Proceedings of Groups' 93 (St. Andrews) London Math. Soc. Lecture Note Series Vol. 212 (Cambridge University Press, 1995), 567–585.Google Scholar