14.104 (1999)

An infinite group $G$ is called a monster of the first kind if it has elements of order $> 2$ and for any such an element $a$ and for any proper subgroup $H$ of $G$, there is an element $g$ in $G \setminus H$, such that $\langle a, a^g \rangle = G$. A. Yu. Olshanskii showed that there are continuously many monsters of the first kind (see 6.63). Does there exist, for any such a monster, a torsion-free group which is a central extension of a cyclic group by the given monster?