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

Document Type : Research Paper

Author

Department of Mathematics, Bu-Ali Sina University, Hamedan, Iran

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 $\lambda\in\{1,2,3,4,6,12‎, ‎16‎, ‎18\}$ and $X=\text{PSU}_n(q)$ with $(n,q)\in\{(2,7),(2,9),(2,11),(3,3),(4,2)\}$‎.

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