Line-Primitive 2-(v, k, 1) Designs with k/(k, v)<=10

A 2&(v, k, 1) design D=(P, L) is a system consisting of a finite set P of v points and a collection L of a k-subset of P, called lines, such that each 2-subset of P is contained in precisely one line. We shall always assume that 2<k<v. Let G Aut(D) be a group of automorphisms of a 2&(v, k, 1) design D. The group G is said to be line-transitive (line-primitive, respectively) on D if G is transitive (primitive, respectively) on L. The group G is said to be point-transitive ( point-primitive, respectively) on D if G is transitive (primitive, respectively) on P. A flag of D is a pair consisting of a point and a line containing this point. G is flag-transitive on D if G is transitive on the set of flags of D. The following results are well known: doi:10.1006 jcta.2000.3079, available online at http: www.idealibrary.com on