10.65 (1986)

Determine the structure of infinite 2-transitive permutation groups $(G, \Omega)$ in which the stabilizer of a point $\alpha \in \Omega$ has the form $G_\alpha = A \cdot G_{\alpha\beta}$ where $G_{\alpha\beta}$ is the stabilizer of two points $\alpha, \beta, \alpha \neq \beta$, such that $G_{\alpha\beta}$ contains an element inverting the subgroup $A$. Suppose, in particular, that $A \setminus \{1\}$ contains at most two conjugacy classes of $G_\alpha$; does $G$ then possess a normal subgroup isomorphic to $PSL_2$ over a field?