A parametrisation for symmetric designs admitting a flag-transitive, point-primitive automorphism group with a product action
Eugenia O’Reilly-Regueiro, José Emanuel Rodríguez-Fitta · Ars Mathematica Contemporanea · 2022
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:D is a design with parameters ((2t+s−1)2, (2t2 − (2−s)t)/s, (t2 − t)/s2)) with s ≥ 1 odd, orG 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 = ml and l ≥ 2 then the complement of D does not admit a flag-transitive automorphism group.