On Flag‐Transitive 2‐(v,k,λ) Designs With λ≤4

ABSTRACT Let be a non‐trivial 2‐ design with , admitting a flag‐transitive automorphism group which is also properly innately transitive on points. Then is of almost simple quotient type. As a corollary, we obtain that is innately transitive on points if and only if it is primitive when . And we proved that there exists a unique 2‐ design for with a automorphism group acting transitively on flags and properly innately transitively on points, that is, the unique symmetric 2‐ design with and its plinth as in [14, Proposition 1.5].