Solvable block transitive automorphism groups of 2−(v,5,1) designs

AbstractLet G be a solvable block transitive automorphism group of a 2−(v,5,1) design and suppose that G is not flag transitive. We will prove that (1)if G is point imprimitive, then v=21, and G⩽Z21:Z6;(2)if G is point primitive, then G⩽AΓL(1,v) and v=pa, where p is a prime number with p≡21(mod40), and a an odd integer