Flag-Transitive \(2\)-\((v, k, 4)\) Symmetric Designs

Abstract

Let \(\mathcal{D}\) be a \(2\)-\((v,k,4)\) symmetric design, and \(G\) be a subgroup of the full automorphism group of \(\mathcal{D}\). In this paper, we prove that if \(G \leq {Aut}(\mathcal{D})\) is flag-transitive, point-primitive then \(G\) is of affine or almost simple type. We prove further that if a nontrivial \(2\)-\((v, k, 4)\) symmetric design has a flag-transitive, point-primitive, almost simple automorphism group \(G\), then \(\text{Soc}(G)\) is not a sporadic simple group.