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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
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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

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