A classification of flag-transitive block designs

TL;DR

This paper classifies certain highly symmetric combinatorial designs with flag-transitive automorphism groups, identifying all possible structures when the automorphism group is almost simple.

Contribution

It provides a complete classification of 2-designs with gcd(r,λ)=1 and flag-transitive automorphism groups, detailing all known examples and infinite families.

Findings

Designs belong to seven infinite families or eleven known examples.

Symmetric designs with gcd(k,λ)=1 are either projective spaces, Hadamard design, or have automorphism groups in AΓL1(q).

Complete description of all such flag-transitive 2-designs.

Abstract

In this article, we investigate $2$-$(v,k,\lambda)$ designs with $\gcd(r,\lambda)=1$ admitting flag-transitive automorphism groups $G$. We prove that if $G$ is an almost simple group, then such a design belongs to one of the seven infinite families of $2$-designs or it is one of the eleven well-known examples. We describe all these examples of designs. We, in particular, prove that if $\mathcal{D}$ is a symmetric $(v,k,\lambda)$ design with $\gcd(k,\lambda)=1$ admitting a flag-transitive automorphism group $G$, then either $G\leq A\Gamma L_{1}(q)$ for some odd prime power $q$, or $\mathcal{D}$ is a projective space or the unique Hadamard design with parameters $(11,5,2)$.