Classification of Flag-Transitive Primitive Symmetric $(v,k,\lambda)$ Designs with $PSL(2,q)$ as Socle

Abstract:

      $\cal D$ be a nontrivial symmetric $(v,k,\lambda)$ design, and $G$ be a subgroup of the full automorphism group of $\cal D$. In this paper we prove that if $G$ acts flag-transitively, point-primitively on $\mathcal{D}$ and ${\rm Soc}(G)={\rm PSL}(2,q)$, then $\mathcal{D}$ has parameters $(7, 3, 1)$, $(7, 4, 2)$, $(11, 5, 2)$, $(11, 6, 3)$ or $(15, 8, 4)$.