14.83 (1999)

We say that an infinite simple group $G$ is a monster of the third kind if for every non-trivial elements $a, b$, of which at least one is not an involution, there are infinitely many elements $g \in G$ such that $\langle a, b^g \rangle = G$. (Compare with V. P. Shunkov’s definitions in Archive, 6.63, 6.64.) Is it true that every simple quasi-Chernikov group is a monster of the third kind? This is true for quasi-finite groups (A. I. Sozutov, Algebra and Logic, 36, no. 5 (1997), 336–348). (We say that a non-$\sigma$ group is quasi-$\sigma$ if all of its proper subgroups have the property $\sigma$.)