18.100 (2014)

For a given class of groups $\mathcal{X}$, let $(\mathcal{X}, \infty)^*$ denote the class of groups in which every infinite subset contains two distinct elements $x$ and $y$ that satisfy $\langle x, x^y \rangle \in \mathcal{X}$. Let $G$ be a finitely generated soluble-by-finite group in the class $(\mathcal{X}, \infty)^*$, let $m$ be a positive integer, and let $\mathcal{F}, \mathcal{E}_m, \mathcal{E}, \mathcal{N}$, and $\mathcal{P}$ denote the classes of finite groups, groups of exponent dividing $m$, groups of finite exponent, nilpotent groups, and polycyclic groups, respectively.
$\qquad$ a) If $\mathcal{X} = \mathcal{E}\mathcal{N}$, then is $G$ in $\mathcal{E}\mathcal{N}$?
$\qquad$ b) If $\mathcal{X} = \mathcal{E}_m\mathcal{N}$, then is $G$ in $\mathcal{E}_m(\mathcal{F}\mathcal{N})$?
$\qquad$ c) If $\mathcal{X} = \mathcal{N}(\mathcal{P}\mathcal{F})$, then is $G$ in $\mathcal{N}(\mathcal{P}\mathcal{F})$?