On finite groups in which commutators are covered by Engel subgroups

Abstract

Abstract Let m , n {m,n} be positive integers and w a multilinear commutator word. Assume that G is a finite group having subgroups G 1 , … , G m {G_{1},\ldots,G_{m}} whose union contains all w-values in G. Assume further that all elements of the subgroups G 1 , … , G m {G_{1},\ldots,G_{m}} are n-Engel in G. It is shown that the verbal subgroup w ⁢ ( G ) {w(G)} is s-Engel for some { m , n , w } {\{m,n,w\}} -bounded number s.