Flag-Transitive Non-Symmetric 2-Designs with $(r,\lambda)=1$ and Alternating Socle

Abstract

This paper deals with flag-transitive non-symmetric 2-designs with $(r,\lambda)=1$. We prove that if $\mathcal D$ is a non-trivial non-symmetric $2$-$(v,k,\lambda)$ design with $(r,\lambda)=1$ and $G\leq Aut(\mathcal D)$ is flag-transitive with $Soc(G)=A_n$ for $n\geq 5$, then $\mathcal D$ is a $2$-$(6,3,2)$ design, the projective space $PG(3,2)$, or a $2$-$(10,6,5)$ design.