Bounded Engel elements in residually finite groups

摘要

Let q be a prime. Let G be a residually finite group satisfying an identity. Suppose that for every x is an element of G there exists a q-power m = m(x) such that the element x(m) is a bounded Engel element. We prove that G is locally virtually nilpotent. Further, let d, n be positive integers and w a non-commutator word. Assume that G is a d-generator residually finite group in which all w-values are n-Engel. We show that the verbal subgroup w(G) has {d, n, w}-bounded nilpotency class.