Flag-transitive 4-(v, k, 4) 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, F. 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 lambda >= 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, 4) designs. Let S = (2, B) be a non-trivial 4 - (q + 1, k, 4) design. If G acts flag-transitively on S, then G is not two-dimensional projective linear group PSL(2, q).