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

TL;DR

This paper classifies symmetric designs with specific automorphism groups, identifying all possible parameters and the corresponding groups for designs with socle as projective special unitary groups of dimension up to five.

Contribution

It provides a complete classification of symmetric designs with flag-transitive, point-primitive automorphism groups whose socle is a small-dimensional projective special unitary group.

Findings

Identified all feasible parameters (v, k, λ) for such designs.

Found exactly eight non-isomorphic designs with specified λ values.

Determined the automorphism groups for each design.

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 $PSU_{3}(3)$, $PSU_{3}(3):2$, $PSU_{4}(2)$ or $PSU_{4}(2):2$.