On flag-transitive 2-(v,k,2) designs

This paper is devoted to the classification of flag-transitive 2-(v,k,2) designs. We show that apart from two known symmetric 2-(16,6,2) designs, every subgroup G automorphism group a nontrivial design primitive affine or almost simple type. Moreover, we classify designs admitting flag transitive with socle PSL(n,q) for some n≥3. Alongside this analysis give construction 2-(v,k−1,k−2) given 2-(v,k,1) which induces 2-transitive action on line. Taking points and lines projective space PG(n−1,3) as input yields G-flag-transitive 2-(v,3,2) where has PSL(n,3) v=(3n−1)/2. Apart these our exactly one other example, namely complement Fano plane.