Flag-transitive block designs and unitary groups

Abstract

In this article, we study $2$-designs with $\gcd (r,\lambda )=1$ admitting a flag-transitive automorphism group. The automorphism groups of these designs are point-primitive of almost simple or affine type. We determine all pairs $(\mathcal{D},G)$, where $\mathcal{D}$ is a $2$-design with $\gcd (r,\lambda )=1$ and $G$ is a flag-transitive almost simple automorphism group of $\mathcal{D}$ whose socle is $X=\mathrm{PSU}(n,q)$ with $(n,q)\ne (3,2)$ and prove that such a design belongs to one of the two infinite families of Hermitian unitals and Witt-Bose-Shrikhande spaces, or it is isomorphic to a design with parameters $(6,3,2)$, $(7,3,1)$, $(8,4,3)$, $(10,6,5)$, $(11,5,2)$ or $(28,7,2)$.

arXiv admin note: text overlap with arXiv:1904.10518