Block-Transitive $2$-$(v,k,1)$ Designs and Groups $E_6(q)$

Abstract:

      This article is a contribution to the study of block-transitive automorphism groups of $2$-$(v, k, 1)$ block designs. Let ${\cal D}$ be a $2$-$(v, k, 1)$ design admitting a block-transitive, point-primitive but not flag-transitive automorphism group $G$. Let $k_r=(k,v-1)$ and $q=p^f$ for prime $p$. In this paper we prove that if $G$ and ${\cal D}$ are as above and $q>$ $(3(k_rk-k_r 1)f)^{1/3}$, then $G$ does not admit a simple group $E_6(q)$ as its socle.