1.90 (1965)

A subgroup $H$ of a group $G$ is called 2-infinitely isolated in $G$ if, whenever the centralizer $C_G(h)$ in $G$ of some element $h \neq 1$ of $H$ contains at least one involution and intersects $H$ in an infinite subgroup, it follows that $C_G(h) \leqslant H$. Let $G$ be an infinite simple locally finite group whose Sylow 2-subgroups are Chernikov groups, and suppose $G$ has a proper 2-infinitely isolated subgroup $H$ containing some Sylow 2-subgroup of $G$. Does it follow that $G$ is isomorphic to a group of the type $\text{PSL}_2(k)$, where $k$ is a field of odd characteristic?