On the Exponent of a Verbal Subgroup in a Finite Group
  Let w be a multilinear commutator word. We prove that if e is a positive integer and G is a finite group in which any nilpotent subgroup generated by w-values has exponent dividing e then the exponent of the corresponding verbal subgroup w(G) is bounded in terms of e and w only. 
Pavel Shumyatsky
math.GR