Two-transitive pairs in $ \mathrm{PSL}(2,q)$

Abstract

For every group $ \mathrm{PSL}(2,q)$, $q$ a prime power, we classify all two-transitive pairs $(U,U_0)$ consisting of a subgroup $U$ of $ \mathrm{PSL}(2,q)$ and a subgroup $U_0$ of $U$ such that the action of $U$ on the cosets of $U_0$ is two-transitive. We obtain twenty classes up to conjugacy in $ \mathrm{PSL}(2,q)$ or fusion in $P\Gamma L (2,q)$ except for two cases in which we don't have that control.