On symmetric designs with flag-transitive and point-quasiprimitive automorphism groups

Abstract

Let $D=\left(P,ℬ\right)$ be a nontrivial symmetric $\left(v,k,\lambda \right)$-design with $\lambda \le 100$, and let $G$ be a flag-transitive automorphism group of $D$. In this paper, we show that if $G$ is quasiprimitive on $P$, then $G$ is of holomorph affine or almost simple type. Moreover, if $G$ is imprimitive on $P$, then $G$ is of almost simple type. According to this observation and to the classification of the finite simple groups we determine all such symmetric designs and the corresponding automorphism groups. We conclude with two open problems and a conjecture.