Abstract: A~$t$-$(v,k,\lambda)$ design is said to be $G$-point-primitive or $G$-block-transitive, if its automorphism group $G$ acts primitively on the point set or transitively on the block set, respectively. In this paper we begin by extending some results on block-transitive Steiner $2$-designs to block-transitive $3$-designs, and then based on these results, investigate the $G$-point-primitive block-transitive $3$-$(v,k,\lambda)$ designs for alternating or symmetric groups $G$. We prove that when $n\geq\min\{\lambda^2, 30\}$ the point stabilizer in $G$ must be of intransitive type, and specifically, when $n\geq30$ there exists no nontrivial $G$-point-primitive block-transitive $3$-$(v,k,2)$ design.

Key words: Block-transitive design, Primitive group, Alternating group