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