Block transitive 2−(v,k,1) designs and PGL2(q) groups

The article mainly deal with the block-transitive automorphism groups of 2 − ( v , k , 1 ) designs. Let G = P G L 2 ( q ) for q a power of an odd prime number. We prove that if G acts block-transitively on a non-trivial 2 − ( v , k , 1 ) design D then D is either a projective plane or 2 − ( 136 , 10 , 1 ) design.