Flag-transitive point-quasiprimitive automorphism groups of 2-designs with λ≤4.

Let D be a non-trivial 2–(v,k,λ) design with λ≤4, and G≤Aut(D) be flag-transitive, point-quasiprimitive, then G is of holomorph affine or almost simple type. As a corollary, we obtain that G cannot be an almost simple group if λ≤3, and also obtain a classification of flag-transitive, point-quasiprimitive and imprimitive 2-(v,k,4) designs.