Inthisarticle,weprovethatifanontrivialsymmetric(v,k,λ)designadmitaflag-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 λ ∈ {1,2,3,4,6,12,16,18} and X = PSUn(q) with (n,q) ∈ {(2,7),(2,9),(2,11),(3,3),(4,2)}.
