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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, everyflag-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 the2-(v,k,2) designs admitting a flag transitive almost simple group G with soclePSL(n,q) for some n \geq 3. Alongside this analysis, we give a construction fora 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 ofpoints and lines of the projective space PG(n-1,3) as input to thisconstruction yields a G-flag-transitive 2-(v,3,2) design where G has soclePSL(n,3) and v=(3^n-1)/2. Apart from these designs, our PSL-classificationyields exactly one other example, namely the complement of the Fano plane.

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