Unsolvable block transitive automorphism groups of \(2-(v,k,1)(k=6,7,8,9)\) designs (Q998443)

Block transitive automorphism groups of Steiner 2-designs of block size \(k\) are studied when \(k \in \{6,7,8,9\}\). When the automorphism group \(G\) is block transitive and point primitive, but not flag transitive, it is shown that if \(G\) is unsolvable, it does not have an exceptional simple group of Lie type as its socle. Moreover when \(k=9\) it is shown that no such group \(G\) that is unsolvable can arise as an automorphism group of a Steiner 2-design.