Unsolvable block transitive automorphism groups of 2-(v,k,1)(k=6,7,8,9) designs

In this paper, we study block transitive automorphism groups of 2-(v,k,1)2-(v,k,1) block designs. Let DD be a 2-(v,k,1)(k=6,7,8,9) design admitting a block transitive, point primitive but not flag transitive group GG of automorphisms. We prove that if GG is unsolvable, then GG does not admit an exceptional simple group of Lie type as its socle. Moreover, for a 2-(v,9,1)2-(v,9,1) design, we also prove that there does not exist any block transitive, point imprimitive, unsolvable group GG of automorphisms.