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

Abstract

For every group PSL(2, q), q a prime power, we classify all two-transitive pairs (U, U 0 ) consisting of a subgroup U of 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-one classes up to conjugacy in PSL(2, q) or fusion in PΓL(2, q) except for two cases in which we don't have that control.