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 $(7,4,2)$, $(11,5,2)$, $(11,6,2)$ or $(45,12,3)$.