Alternating Groups and Flag-Transitive 2-(v, k, 4) Symmetric Designs

In this article, we study the classification of flag-transitive, point-primitive 2-(v, k, 4) symmetric designs. We prove that if the socle of the automorphism group G of a flag-transitive, point-primitive nontrivial 2-(v, k, 4) symmetric design D is an alternating group An for n >= 5, then (v, k)=(15,8) and D is one of the following: (i) The points of D are those of the projective space PG(3,2) and the blocks are the complements of the planes of PG(3,2), G= A(7) or A(8), and the stabilizer G(x) of a point x of D is L(3)(2) or AGL(3)(2), respectively. (ii) The points of D are the edges of the complete graph K(6) and the blocks are the complete bipartite subgraphs K(2,4) of K(6), G= A(6) or S(6), and G(x) = S(4) or S(4) x Z(2), respectively. (C) 2011 Wiley Periodicals, Inc. J Combin Designs 19: 475-483, 2011