Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-24T02:27:43.319Z Has data issue: false hasContentIssue false
  • English
  • Français

On locally graded groups with a word whose values are Engel

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  17 December 2015

Pavel Shumyatsky ,
Antonio Tortora  and
Maria Tota
Pavel Shumyatsky
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia, Federal District 70910-900, Brazil ([email protected])
Antonio Tortora
Affiliation:
Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, 132-84084, Fisciano, Salerno, Italy ([email protected]; [email protected])
Maria Tota
Affiliation:
Dipartimento di Matematica, Università di Salerno, Via Giovanni Paolo II, 132-84084, Fisciano, Salerno, Italy ([email protected]; [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let m, n be positive integers, let υ be a multilinear commutator word and let w = υm. We prove that if G is a locally graded group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent.

Keywords

Engel elementsmultilinear commutatorslocally graded groups

MSC classification

Primary: 20F45: Engel conditions
Secondary: 20F40: Associated Lie structures
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 59 , Issue 2 , May 2016 , pp. 533 - 539
Copyright
Copyright © Edinburgh Mathematical Society 2016 

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

1. Bastos, R., Shumyatsky, P., Tortora, A. and Tota, M., On groups admitting a word whose values are Engel, Int. J. Alg. Comput. 23 (2013), 81–89.Google Scholar
2. Hall, M. Jr, The theory of groups (Macmillan, London, 1959).Google Scholar
3. Kim, Y. and Rhemtulla, A. H., On locally graded groups, in Groups–Korea ’94, pp. 189–197 (Walter de Gruyter, Berlin, 1995).CrossRefGoogle Scholar
4. Longobardi, P., Maj, M. and Smith, H., A note on locally graded groups, Rend. Sem. Mat. Univ. Padova 94 (1995), 275–277.Google Scholar
5. Plotkin, B. I., Radicals and nil-elements in groups, Izv. VUZ Mat. 1 (1958), 130–135.Google Scholar
6. Robinson, D. J. S., A course in the theory of groups, 2nd edn (Springer, 1996).Google Scholar
7. Shumyatsky, P., On residually finite groups in which commutators are Engel, Commun. Alg. 27 (1999), 1937–1940.CrossRefGoogle Scholar
8. Shumyatsky, P., Applications of Lie ring methods to group theory, in Nonassociative algebra and its applications, Lecture Notes in Pure and Applied Mathematics, Volume 211, pp. 373–395 (Dekker, New York, 2000).Google Scholar
9. Shumyatsky, P., Verbal subgroups in residually finite groups, Q. J. Math. 51 (2000), 523–528.CrossRefGoogle Scholar
10. Shumyatsky, P., On varieties arising from the solution of the restricted Burnside problem, J. Pure Appl. Alg. 171(1) (2002), 67–74.Google Scholar
11. Shumyatsky, P., Elements of prime power order in residually finite groups, Int. J. Alg. Comput. 15(3) (2005), 571–576.Google Scholar
12. Wilson, J. S., Two-generator conditions for residually finite groups, Bull. Lond. Math. Soc. 23 (1991), 239–248.CrossRefGoogle Scholar
13. Zelmanov, E. I., Solution of the restricted Burnside problem for groups of odd exponent, Math. USSR Izv. 36(1) (1991), 41–60.Google Scholar
14. Zelmanov, E. I., Solution of the restricted Burnside problem for 2-groups, Math. USSR Sb. 72(2) (1992), 543–565.CrossRefGoogle Scholar