Block designs with $\gcd (r,\lambda )=1$ admitting flag-transitive automorphism groups

Abstract

In this paper, we present a classification of $2$-designs with $\gcd (r,\lambda )=1$ admitting flag-transitive automorphism groups. If $G$ is a flag-transitive automorphism group of a non-trivial $2$-design $\mathcal{D}$ with $\gcd (r,\lambda )=1$, then either $(\mathcal{D},G)$ is one of the known examples described in this paper, or $\mathcal{D}$ has $q=p^d$ points with $p$ prime and $G$ is a subgroup of $A\Gamma L_1(q)$.