Flag-transitive $2$-$(v,k,\lambda)$ designs with $\lambda\ge (r,\lambda)^2$

TL;DR

This paper classifies certain flag-transitive 2-designs with specific parameters, focusing on those with automorphism groups of almost simple type involving projective special linear or unitary groups.

Contribution

It provides a classification of 2-designs with λ ≥ (r,λ)^2 > 1 admitting flag-transitive automorphism groups of almost simple type with socles PSL_n(q) or PSU_n(q).

Findings

Classified 2-designs with specified λ and automorphism groups.

Identified conditions for flag-transitivity in these designs.

Extended understanding of symmetry properties in combinatorial designs.

Abstract

This paper is devoted to the study of $2$-designs with $\lambda\ge (r,\lambda)^2$ admitting a flag-transitive automorphism group $G$. The group $G$ has been shown to be point-primitive of either almost simple or affine type. In this paper, we classify the $2$-designs with $\lambda \geq (r,\lambda)^2>1$ admitting a flag-transitive almost simple automorphism group with socle $\mathrm{PSL}_n(q)$ or $\mathrm{PSU}_n(q)$ for $n \geq 3$.