Some properties of ultra Hall and Schur pairs

Abstract.

Let ${\cal X}$ be a class of groups satisfying ${\cal X} = PQS_n {\cal X}$, then a necessary and sufficient condition is given for $(\Phi, {\cal X})$ to have the Schur pair property, for every set of words ${\Phi}$. We also introduce the notion of ultra Hall pair and show that $(\phi, {\cal X})$ is an ultra Hall pair for every outer commutator word ${\phi}$, when $(\gamma_2, {\cal X})$ is a Schur pair. Finally, a generalized version of Hall's second problem is given for the class of ${\cal X}$.