Khukhro, Evgeny and Shumyatsky, Pavel (2019) Compact groups in which all elements are almost right Engel. Quarterly Journal of Mathematics . ISSN 00335606
Full content URL: https://doi.org/10.1093/qmath/haz002
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Item Type:  Article 

Item Status:  Live Archive 
Abstract
We say that an element g of a group G is almost right Engel if there is a finite set R(g) such that for every x∈G all sufficiently long commutators [...[[g,x],x],…,x] belong to R(g), that is, for every x∈G there is a positive integer n(x,g) such that [...[[g,x],x],…,x]∈R(g) if x is repeated at least n(x,g) times. Thus, g is a right Engel element precisely when we can choose R(g)={1}. We prove that if all elements of a compact (Hausdorff) group G are almost right Engel, then G has a finite normal subgroup N such that G/N is locally nilpotent. If in addition there is a uniform bound R(g)≤m for the orders of the corresponding sets, then the subgroup N can be chosen of order bounded in terms of m. The proofs use the WilsonZelmanov theorem saying that Engel profinite groups are locally nilpotent and previous results of the authors about compact groups all elements of which are almost left Engel.
Keywords:  compact group, profinite group, finite group, Engel condition, locally nilpotent group 

Subjects:  G Mathematical and Computer Sciences > G110 Pure Mathematics 
Divisions:  College of Science > School of Mathematics and Physics 
ID Code:  34681 
Deposited On:  01 Feb 2019 13:23 
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