Khukhro, Evgeny and Shumyatsky, Pavel
(2018)
Almost Engel compact groups.
Journal of Algebra, 500
.
pp. 439456.
ISSN 00218693
Full content URL: https://doi.org/10.1016/j.jalgebra.2017.04.021
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Abstract
We say that a group G is almost Engel if for every g∈G there is a finite set E(g) such that for every x∈G all sufficiently long commutators [...[[x,g],g],…,g] belong to E(g), that is, for every x∈G there is a positive integer n(x,g) such that [...[[x,g],g],…,g]∈E(g) if g is repeated at least n(x,g) times. (Thus, Engel groups are precisely the almost Engel groups for which we can choose E(g)={1} for all g∈G.) We prove that if a compact (Hausdorff) group G is almost Engel, then G has a finite normal subgroup N such that G/N is locally nilpotent. If in addition there is a unform bound E(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 Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent.
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