On profinite groups with Engel-like conditions

Let G be a profinite group in which for every element x is an element of G there exists a natural number q = q(x) such that x(q) is Engel. We show that G is locally virtually nilpotent. Further, let p be a prime and G a finitely generated profinite group in which for every gamma kappa-value x is an element of G there exists a natural p-power q = q(x) such that x(q) is Engel. We show that gamma kappa(G) is locally virtually nilpotent. (c) 2015 Elsevier Inc. All rights reserved.