The Higman-M\lowercase{c}Laughlin Theorem for the flag-transitive $2$-designs with $\lambda$ prime

Abstract

A famous result of Higman and McLaughlin \cite{HM} in 1961 asserts that any flag-transitive automorphism group $G$ of a $2$-design $\mathcal{D}$ with $\lambda=1$ acts point-primitively on $\mathcal{D}$. In this paper, we show that the Higman and McLaughlin theorem is still true when $\lambda$ is a prime and $\mathcal{D}$ is not isomorphic to one of the two $2$-$(16,6,2)$ designs as in [42, Section 1.2], or the $2$-$(45,12,3)$ design as in [44, Construction 4.2], or, when $2^{2^{j}}+1$ is a Fermat prime, a possible $2$-$(2^{2^{j+1}}(2^{2^{j}}+2),2^{2^{j}}(2^{2^{j}}+1),2^{2^{j}}+1)$ design having very specific features.