Flag-Transitive Non-Symmetric 2-Designs with $(r, \lambda)=1$ and Exceptional Groups of Lie Type

Abstract

This paper  determines all  pairs $(\mathcal{D},G)$ where $\mathcal{D}$ is a non-symmetric 2-$(v,k,\lambda)$ design   with $(r,\lambda)=1$ and  $G$ is  the  almost simple flag-transitive automorphism group of $\mathcal{D}$ with  an exceptional  socle of Lie type. We prove that if $T\trianglelefteq G\leq Aut(T)$ where $T$ is an exceptional group of Lie type, then $T$ must be the Ree group or Suzuki group, and there are five classes of designs $\mathcal{D}$.