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

TL;DR

This paper classifies certain 2-designs with block size seven that admit flag-transitive automorphism groups with alternating groups as socles, identifying specific known designs and their automorphism groups.

Contribution

It provides a complete classification of 2-designs with block size seven under flag-transitive automorphism groups with alternating socles, identifying only two specific designs.

Findings

Design with parameters (15,7,3) and automorphism group A7

Design with parameters (55,7,1680) and automorphism group A11 or S11

Complete classification of such 2-designs with block size 7

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 $S_{11}$.