Flag-Transitive Point-Primitive Symmetric $(v,k,\lambda)$ Designs with Large $k$

Abstract

In 2012, Tian and Zhou conjectured that a flag-transitive and point-primitive automorphism group of a symmetric $(v,k,\lambda)$ design must be an affine or almost simple group. In this paper, we study this conjecture and prove that if $k\leqslant 10^3$ and $G\leqslant Aut(\mathcal{D})$ is flag-transitive, point-primitive, then $G$ is affine or almost simple. This support the conjecture.