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FINITE GROUPS WITH ENGEL SINKS OF BOUNDED RANK

Part of: Representation theory of groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  28 January 2018

E. I. KHUKHRO Open the ORCID record for E. I. KHUKHRO [Opens in a new window]  and
P. SHUMYATSKY
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
Rights & Permissions [Opens in a new window]

Abstract

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For an element g of a group G, an Engel sink is a subset ${\mathscr E}$(g) such that for every x ∈ G 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.

MSC classification

Primary: 20F45: Engel conditions
Secondary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks 20D06: Simple groups
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 3 , September 2018 , pp. 695 - 701
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

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), 520–534.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), 161–170; 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), 17–22; English transl., Math. USSR Dokl. 5 (1964), 591–594.Google Scholar
4. Guralnick, R. M., Generation of simple groups, J. Algebra 103 (1986), 381–401.CrossRefGoogle Scholar
5. Guralnick, R. M., On the number of generators of a finite group, Arch. Math. (Basel) 53 (1989), 521–523.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), 1–42.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), 153–173.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), 474–492.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), 37–39.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), 201–207.CrossRefGoogle Scholar
12. Longobardi, P. and Maj, M., On the number of generators of a finite group, Arch. Math. (Basel) 50 (1988), 110–112.CrossRefGoogle Scholar
13. Lucchini, A., A bound on the number of generators of a finite group, Arch. Math. (Basel) 53 (1989), 313–317.CrossRefGoogle Scholar
14. Medvedev, Yu., On compact Engel groups, Isr. J. Math. 185 (2003), 147–156.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), 402–412.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), 103–109.CrossRefGoogle Scholar
19. Zorn, M., Nilpotency of finite groups, Bull. Amer. Math. Soc. 42 (1936), 485–486.Google Scholar