Symmetric designs and projective special unitary groups of dimension at most five

Abstract

In this article, we study symmetric $(v,k,\lambda )$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle is a projective special unitary group of dimension at most five. We, in particular, determine all such possible parameters $(v,k,\lambda )$ and show that there exist eight non-isomorphic of such designs for which $\lambda \in \{3,6,12,16,18\}$ and $G$ is $PS{U}_3(3)$, $PS{U}_3(3):2$, $PS{U}_4(2)$ or $PS{U}_4(2):2$.