Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T04:35:42.341Z Has data issue: false hasContentIssue false
  • English
  • Français

On profinite groups in which commutators are Engel

Part of: Special aspects of infinite or finite groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Pavel Shumyatsky
Pavel Shumyatsky
Affiliation:
Department of Mathematics University of Brasilia70910—900 Brasilia DFBrazil e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that if G is a finitely generated profinite group such that [x1, x2, …, xk] is Engel for any x1, x2, …, xk ∈ G, then γ(G) is locally nilpotent, and if [x1, x2, …, xk] has finite order for any x1, x2, …, xk ∈ G then, under some additional assumptions, γk(G) is locally finite.

Keywords

profinite groupsassociated Lie algebra

MSC classification

Secondary: 20E18: Limits, profinite groups 20F40: Associated Lie structures 20F45: Engel conditions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 70 , Issue 1 , February 2001 , pp. 1 - 9
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Dixon, J. D., du Sautoy, M. P. F., Mann, A. and Segal, D., Analytic p-adic groups (Cambridge University Press, Cambridge, 1991).Google Scholar
[2]Hartley, B., ‘Subgroups of finite index in profinite groups’, Math. Z. 168 (1979), 71–76.CrossRefGoogle Scholar
[3]Huppert, B., Endliche Gruppen I (Springer, Berlin, 1967).CrossRefGoogle Scholar
[4]Kelley, J. L., General topology (Van Nostrand, New York, 1955).Google Scholar
[5]Lazard, M., ‘Sur les groupes nilpotents et les anneaux de Lie’, Ann. Sci. École Norm. Supr. 71 (1954), 101–190.CrossRefGoogle Scholar
[6]Lazard, M., ‘Groupes analytiques p-adiques’, Publ. Math. Inst. Hautes Études Sci. 26 (1965), 389–603.Google Scholar
[7]Robinson, D. J. S., Finiteness conditions and generalized soluble groups (Springer, Berlin, 1972).CrossRefGoogle Scholar
[8]Shumyatsky, P., ‘On groups with commutators of bounded order’, Proc. Amer Math. Soc. 127 (1999), 2583–2586.CrossRefGoogle Scholar
[9]Shumyatsky, P., ‘Residually finite groups in which commutators are Engel’, Comm. Algebra 27 (1999), 1937–1940.CrossRefGoogle Scholar
[10]Shumyatsky, P., ‘Applications of Lie ring methods to group theory’, in: Nonassociative algebra and its applications (eds. Costa, R., Grishkov, A., Guzzo, H. Jr and Peresi, L. A.), Lecture Notes in Pure and Appl. Math. (Marcel Dekker, New York, 2000) pp. 373–395.Google Scholar
[11]Tits, J., ‘Free subgroups in linear groups’, J. Algebra 20 (1972), 250–270.CrossRefGoogle Scholar
[12]Wilson, J. S., ‘On the structure of compact torsion groups’, Monatshefte für Mathematik 96 (1983), 57–66.CrossRefGoogle Scholar
[13]Wilson, J. S., ‘Two-generator conditions for residually finite groups’, Bull. London Math. Soc. 23 (1991), 239–248.CrossRefGoogle Scholar
[14]Wilson, J. S. and Zel'manov, E. I., ‘Identities for Lie algebras of pro-p groups’, J. Pure Appl. Algebra 81 (1992), 103–109.CrossRefGoogle Scholar
[15]Zelmanov, E. I., ‘The solution of the restricted Burnside problem for groups of odd exponent’, Math. USSR Izv. 36 (1991), 41–60.CrossRefGoogle Scholar
[16]Zelmanov, E. I., ‘The solution of the restricted Burnside problem for 2-groups’, Math. Sb. 182 (1991), 568–592.Google Scholar
[17]Zelmanov, E. I., ‘On periodic compact groups’, Israel J. Math. 77 (1992), 83–95.CrossRefGoogle Scholar
[18]Zelmanov, E. I., Nil Rings and Periodic Groups, The Korean Math. Soc. Lecture Notes in Math. (Seoul, 1992).Google Scholar