Some remarks on generalized Schur pairs

In 1940, P. Hall introduced the concept of a Schur pair property, and J. A. Hulse and J. C. Lennox (1976) generalized this notion to an ultra Schur pair. In this paper we introduce the notion of an ultra weak schur pair and show that if X is a class of groups such that X = NPQS(n)X and (gamma(2), X) is a Schur pair, then (phi, X) is an ultra weak Schur pair, for every outer commutator word phi. Finally, we deal with P. Hall's second problem.