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 $(n,q)in{(2,7),(2,9),(2,11),(3,3),(4,2)}$.
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