Block-transitive $3$-$(v,k,1)$ designs associated with alternating groups

Abstract

Let $\mathcal{D}$ be a nontrivial $3$-$(v,k,1)$ design admitting a block-transitive group $G$ of automorphisms. A recent work of Gan and the second author asserts that $G$ is either affine or almost simple. In this paper, it is proved that if $G$ is almost simple with socle an alternating group, then $\mathcal{D}$ is the unique $3$-$(10,4,1)$ design, and $G=\mathrm{PGL}(2,9)$, ${\mathrm{M}}_{10}$ or $\mathrm{Aut}({\mathrm{A}}_6)={\mathrm{S}}_6:{\mathrm{Z}}_2$, and $G$ is flag-transitive.