On Flag-Transitive, Point-Quasiprimitive Symmetric 2-(v,k,) Designs with Prime

This paper contributes to the classification of flag-transitive symmetric 2-(v,k,lambda) designs with lambda prime. We investigate the structure of flag-transitive, point-quasiprimitive automorphism groups (G) of such 2-designs by applying the classification of quasiprimitive permutation groups. It is shown that the automorphism groups (G) have either an abelian socle or a non-abelian simple socle. Moreover, according to the classification of finite simple groups, we demonstrate that point-quasiprimitivity implies point-primitivity of G, except when the socle of G is PSLn(q).