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 is an alternating group A n for n ≥5, then ( v , k ) = (15, 8) and is one of the following: (i) The points of 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 is L 3 (2) or AGL 3 (2), respectively. (ii) The points of 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 × Z 2 , respectively. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:475‐483, 2011