Abstract
Let m, n be positive integers. Suppose that G is a residually finite group in which for every element x ∈ G there exists a positive integer q = q(x) ≤ m such that xq is left n-Engel. We show that G is locally virtually nilpotent. Further, let w be a multilinear commutator and G a residually finite group in which for every product of at most 896 w-values x there exists a positive integer q = q(x) dividing m such that xq is left n-Engel. Then w(G) is locally virtually nilpotent.
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