Almost Simple Groups of Lie Type and Symmetric Designs with $\lambda$ Prime

Abstract

In this article, we investigate symmetric $(v,k,\lambda)$ designs $\mathcal{D}$ with $\lambda$ prime admitting flag-transitive and point-primitive automorphism groups $G$. We prove that if $G$ is an almost simple group with socle a finite simple group of Lie type, then $\mathcal{D}$ is either the point-hyperplane design of a projective space $\mathrm{PG}_{n-1}(q)$, or it is of parameters  $(7,4,2)$, $(11,5,2)$, $(11,6,3)$ or $(45,12,3)$.