Flag-transitive 4-(v, k, 3) designs and PSL(2, q) groups

Among the properties of homogeneity of incidence structures, flag-transitivity obviously is a particularly important and natural one. Originally, Buekenhout et al. reached a classification of flag-transitive Steiner 2-designs. Recently, Huber completely classified all flag-transitive Steiner t -designs with t ≤ 6 using the classification of the finite 2-transitive permutation groups. Hence the determination of all flag-transitive t -designs with λ ≥ 2 has remained of particular interest and has been known as a long-standing and still open problem.This article is a contribution to the study of the automorphism groups of 4-( v, k , 3) designs. Let S = ( P , B ) be a non-trivial 4- ( q + 1 , k , 3 ) design. If PSL (2, q ) acts flag-transitively on S , then S is a 4-(168,12,3) design and G B is conjugate to A 4 or Z 12 .