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

Abstract

We study (v,k,λ)-symmetric designs having a flag-transitive, point-primitive automorphism group, with v = m² and (k,λ) = 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 ((2t+s−1)², (2t² − (2−s)t)/s, (t² − t)/s²)) 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,λ)-symmetric design with a flag-transitive, point-primitive 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.