On 2‐Designs Whose Group of Automorphisms is Transitive on the Ordered Pairs of Intersecting Lines

THEOREM 1. Let G be a permutation group on a finite set S of v points. If a, b are distinct points then the set ab of all points fixed by the stabilizer Gab will be called aline. Assume that (i) G is 2-transitive on the points of S; (ii) the number k of points on the line ab satisfies 2 < k < v; (iii) Gab is transitive on the lines containing a and distinct from ab. Then S provided with the set of lines is an affine space AG(d, q) of dimension d^lon some field oforderq ^ 3 or Sis a projective space PG(d, 2) of dimension d ^ 2 and order 2.