Unsolvable Block-Transitive Automorphism Groups of $2–(v,31,1)$ Designs

Abstract

This paper contributes to the study of automorphism groups of $2-(v,k,1)$ designs. Let $\mathcal{D}$ be a $2-(v,31,1)$ design and $G\le Aut(\mathcal{D})$ be block-transitive and point-primitive. If $G$ is unsolvable, then $Soc(G)$, the socle of $G$, is not isomorphic to ${}^2{F}_4(q)$.