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 $G\le Aut(\mathcal{D})$ 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.