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 $G\le A\Gamma L_1(q)$ for some odd prime power $q$, or $\mathcal{D}$ is a projective space or the unique Hadamard design with parameters $(11,5,2)$.

arXiv admin note: text overlap with arXiv:1904.10518