Two-transitive pairs in \(\mathrm{PSL}(2,q)\) (Q267509)

This paper is devoted to the classification of all 2-transitive pairs of a group \(G=\mathrm{PSL}(2,q)\), \(q\) a prime power, namely for every group \(\mathrm{PSL}(2,q)\), they 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. They obtain twenty-one 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.