Reduction for flag-transitive symmetric designs with $k>\lambda (\lambda -2)$

Let $G$ be a flag-transitive automorphism group of a $(v,k,\lambda )$ symmetric design $\mathcal{D}$ with $k>\lambda (\lambda -2)$. O’Reilly Regueiro proved that if $G$ is point-imprimitive, then $\mathcal{D}$ has parameters $(v,k,\lambda )=({\lambda }^2(\lambda +2),\lambda (\lambda +1),\lambda )$. In the present paper, we consider the case that $G$ is point-primitive. By applying the O’Nan-Scott Theorem, we prove that $G$ must be of affine type or almost simple type.