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

This article is a contribution to the study of the automorphism groups of 2-(v,k,1) block designs. In particular we look at the unsolvable block transitive automorphism groups of 2-(v,5,1) designs and prove the following theorem.Main TheoremIf a block transitive group of automorphisms of a 2-(v,5,1) design is unsolvable, then it is flag transitive.