Skip to main content Accessibility help
×
Home
Hostname: page-component-78dcdb465f-8p2q5 Total loading time: 0.229 Render date: 2021-04-19T05:52:33.247Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

FINITE GROUPS WITH ENGEL SINKS OF BOUNDED RANK

Published online by Cambridge University Press:  28 January 2018

E. I. KHUKHRO
Affiliation:
University of Lincoln, Lincoln, LN6 7TS, United Kingdom Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia e-mail: khukhro@yahoo.co.uk
P. SHUMYATSKY
Affiliation:
Department of Mathematics, University of Brasilia, DF 70910-900, Brazil e-mail: pavel@unb.br
Corresponding

Abstract

For an element g of a group G, an Engel sink is a subset ${\mathscr E}$(g) such that for every xG all sufficiently long commutators [. . .[[x, g], g], . . ., g] belong to ${\mathscr E}$(g). A finite group is nilpotent if and only if every element has a trivial Engel sink. We prove that if in a finite group G every element has an Engel sink generating a subgroup of rank r, then G has a normal subgroup N of rank bounded in terms of r such that G/N is nilpotent.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below.

References

1. Berger, T. R. and Gross, F., 2-length and the derived length of a Sylow 2-subgroup, Proc. Lond. Math. Soc. 34 (3) (1977), 520534.CrossRefGoogle Scholar
2. Bryukhanova, E. G., The relation between 2-length and derived length of a Sylow 2-subgroup of a finite solvable group, Mat. Zametki 29 (2) (1981), 161170; English transl., Math. Notes 29(1–2) (1981), 85–90.Google Scholar
3. Gorchakov, Yu. M., On existence of abelian subgroups of infinite ranks in locally solvable groups, Dokl. Akad. Nauk SSSR 146 (1964), 1722; English transl., Math. USSR Dokl. 5 (1964), 591–594.Google Scholar
4. Guralnick, R. M., Generation of simple groups, J. Algebra 103 (1986), 381401.CrossRefGoogle Scholar
5. Guralnick, R. M., On the number of generators of a finite group, Arch. Math. (Basel) 53 (1989), 521523.CrossRefGoogle Scholar
6. Hall, P. and Higman, G., The p-length of p-soluble groups and reduction theorems for Burnside's problem, Proc. Lond. Math. Soc. 6 (3) (1956), 142.CrossRefGoogle Scholar
7. King, C. S. H., Generation of finite simple groups by an involution and an element of prime order, J. Algebra 478 (2017), 153173.CrossRefGoogle Scholar
8. Khukhro, E. I. and Mazurov, V. D., Finite groups with an automorphism of prime order whose centralizer has small rank, J. Algebra 301 (2006), 474492.CrossRefGoogle Scholar
9. Khukhro, E. I. and Shumyatsky, P., Almost Engel compact groups, J. Algebra, to appear, doi 10.1016/j.jalgebra.2017.04.021; http://arxiv.org/abs/1610.02079.Google Scholar
10. Kovács, L. G., On finite soluble groups, Math. Z. 103 (1968), 3739.CrossRefGoogle Scholar
11. Liebeck, M. W., Nikolov, N. and Shalev, A., Groups of Lie type as products of SL 2 subgroups J. Algebra 326 (2011), 201207.CrossRefGoogle Scholar
12. Longobardi, P. and Maj, M., On the number of generators of a finite group, Arch. Math. (Basel) 50 (1988), 110112.CrossRefGoogle Scholar
13. Lucchini, A., A bound on the number of generators of a finite group, Arch. Math. (Basel) 53 (1989), 313317.CrossRefGoogle Scholar
14. Medvedev, Yu., On compact Engel groups, Isr. J. Math. 185 (2003), 147156.CrossRefGoogle Scholar
15. Merzlyakov, Yu. I., On locally soluble groups of finite rank, Algebra Log. 3 (2) (1964), 5–16 (in Russian).Google Scholar
16. Robinson, D. J. S., A course in the theory of groups (Springer, New York, 1996).CrossRefGoogle Scholar
17. Roseblade, J. E., On groups in which every subgroup is subnormal, J. Algebra 2 (1965), 402412.CrossRefGoogle Scholar
18. Wilson, J. S. and Zelmanov, E. I., Identities for Lie algebras of pro-p groups, J. Pure Appl. Algebra 81 (1) (1992), 103109.CrossRefGoogle Scholar
19. Zorn, M., Nilpotency of finite groups, Bull. Amer. Math. Soc. 42 (1936), 485486.Google Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 56 *
View data table for this chart

* Views captured on Cambridge Core between 28th January 2018 - 19th April 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

FINITE GROUPS WITH ENGEL SINKS OF BOUNDED RANK
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

FINITE GROUPS WITH ENGEL SINKS OF BOUNDED RANK
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

FINITE GROUPS WITH ENGEL SINKS OF BOUNDED RANK
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *