On flag-transitive symmetric (v, k, 4) designs

The article explores symmetric designs with small admitting flag-transitive automorphism groups, building upon foundational work by Higman, McLaughlin, and Kantor. It presents a thorough analysis of flag-transitive point-imprimitive and point-primitive symmetric designs, with a particular focus on the affine type automorphism groups. The main result, Theorem 1.1, provides a classification of nontrivial symmetric designs with parameter set (v, k, 4) admitting a flag-transitive and point-primitive automorphism group of affine type. The article also offers a solution to Problem 1 in [29], reducing the classification of flag-transitive symmetric (v, k, 4) designs to the case of one-dimensional affine automorphism groups. Additionally, it presents a classification of symmetric designs with λ = 3 or 4 admitting flag-transitive automorphism groups, listing these designs in Table 1. The article employs advanced mathematical techniques, including the O’Nan-Scott Theorem and Aschbacher’s Theorem, to analyze the automorphism groups and their actions on the designs. It also discusses the geometric and algebraic structures underlying these designs, providing insights into their properties and relationships.