A parametrisation for symmetric designs admitting a flag-transitive, point-primitive automorphism group with a product action

We study (v, k, lambda)-symmetric designs having a flag-transitive, point-primitive automorphism group, with v = m(2) and (k, lambda) = t > 1, and prove that if D is such a design with m even admitting a flag-transitive, point-primitive automorphism group G, then either: (1) D is a design with parameters (1) D (2t + s - 1)(2), 2t(2)-(2-s)t/s, t(2)-t/s(2)) with s >= 1 odd, or (2) G does not have a non-trivial product action. We observe that the parameters in (1), when s = 1, correspond to Menon designs. We also prove that if D is a (v, k, lambda)-symmetric design with a flag-transitive, pointprimitive automorphism group of product action type with v = m(l) and l >= 2 then the complement of D does not admit a flag-transitive automorphism group.