6.64 (1978)
SolvedA group $G$ is called a monster of the second kind if it has elements of order $> 2$ and if for any such element $a$ and any proper subgroup $H$ of $G$ there exists an infinite subset $\mathcal{M}_{a,H}$ consisting of conjugates of $a$ by elements of $G \setminus H$ such that $\langle a, c \rangle = G$ for all $c \in \mathcal{M}_{a,H}$. Do mixed monsters (that is, with elements of both finite and infinite orders) of the second kind exist? Do there exist torsion-free monsters of the second kind?
Progress
Yes, such monsters exist, in both cases (A. Yu. Olshanskii, The geometry of defining relations in groups, Kluwer, Dordrecht, 1991).
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