Symmetric designs and projective special unitary groups $text{PSU}_{5}(q)$

Abstract

In this article‎, ‎we prove that if a nontrivial symmetric $(v‎, ‎k‎, ‎lambda)$ design admit a flag-transitive and point-primitive automorphism group $G$‎, ‎then the socle $X$ of $G$ cannot be a projective special unitary group of dimension five‎. ‎As a corollary‎, ‎we list all exist nineteen non-isomorphism such designs in which $lambdain{1,2,3,4,6,12‎, ‎16‎, ‎18}$ and $X=text{PSU}_n(q)$ with $(n,q)in{(2,7),(2,9),(2,11),(3,3),(4,2)}$‎.