Flag-transitive primitive (v, k, λ) symmetric designs with λ at most 10 and alternating socle

This paper is a contribution to the investigation of flag-transitive primitive symmetric (v, k, lambda) designs with lambda at most 10. We prove that if D = (P, B) is a nontrivial symmetric (v, k, lambda) design, G <= Aut(D) is flag-transitive primitive with Soc(G) = A(n) for n >= 5, then D is the unique symmetric (35, 18, 9) design which is the complement of the symmetric (35, 17, 8) design.