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 GB is conjugate to A4 or Z12.