Block-Transitive Steiner 3-Designs on 2-Transitive Groups

In 2005, Huber proved that the flag-transitive group of automorphisms for a Steiner Formula: see text-design must act Formula: see text-transitively on the point set. He also classified flag-transitive Steiner Formula: see text-designs. It should be noted that while flag-transitive designs are block-transitive, the converse is not always true. Furthermore, all known block-transitive Steiner Formula: see text-designs have Formula: see text-transitive automorphism groups. In this study, we consider Formula: see text-block-transitive Steiner Formula: see text-designs Formula: see text with Formula: see text acting Formula: see text-transitively on the point set, and prove that if Formula: see text is not Formula: see text-flag-transitive, then either Formula: see text is a Formula: see text-Formula: see text-design with the Suzuki group as its automorphism group, or Formula: see text is a subgroup of Formula: see text and the size of the block is an odd integer.