19.1 (2018)

An element $g$ of a group $G$ is a non-near generator of $G$ if for every subset $S \subseteq G$ such that $|G : \langle g, S \rangle| < \infty$ it follows that $|G : \langle S \rangle| < \infty$. The set of all non-near generators of $G$ forms a characteristic subgroup of $G$ called the lower near Frattini subgroup of $G$, denoted by $\lambda(G)$. A subgroup $M \leqslant G$ is nearly maximal in $G$ if it is maximal with respect to being of infinite index in $G$. The intersection of all nearly maximal subgroups forms a characteristic subgroup called the upper near Frattini subgroup of $G$, denoted by $\mu(G)$. In general, $\lambda(G) \leqslant \mu(G)$. If $\lambda(G) = \mu(G)$, then this subgroup is called the near Frattini subgroup of $G$, denoted by $\psi(G)$.
$\qquad$ a) Is it true that $\psi(G) = 1$ if $G$ is the knot group of any product of knots?
$\qquad$ b) Is it true that $\psi(G) = 1$ if $G$ is a cable knot group?