Alternating groups as flag-transitive automorphism groups of 2-designs with block size seven

Abstract

In this article, we prove that if $\mathcal{D}$ is a $2$-design with $k=7$ admitting flag-transitive almost simple automorphism group with socle an alternating group, then $\mathcal{D}$ is $PG_{2}(3,2)$ with parameter set $(15,7,3)$ and $G=A_7$, or $\mathcal{D}$ is the $2$-design with parameter set $(55, 7, 1680)$ and $G=A_{11}$ or $S_{11}$.