A classification of flag-transitive block designs

Abstract

In this article, we investigate $2$-$(v,k,\\lambda)$ designs with $\\gcd(r,\\lambda)=1$ admitting flag-transitive automorphism groups $G$. We prove that if $G$ is an almost simple group, then such a design belongs to one of the seven infinite families of $2$-designs or it is one of the eleven well-known examples. We describe all these examples of designs. We, in particular, prove that if $\\mathcal{D}$ is a symmetric $(v,k,\\lambda)$ design with $\\gcd(k,\\lambda)=1$ admitting a flag-transitive automorphism group $G$, then either $G\\leq A\\Gamma L_{1}(q)$ for some odd prime power $q$, or $\\mathcal{D}$ is a projective space or the unique Hadamard design with parameters $(11,5,2)$.