The Higman-M\lowercase{c}Laughlin Theorem for the flag-transitive $2$-designs with $λ$ prime
Alessandro Montinaro
TL;DR
This work extends the Higman–McLaughlin theorem to flag-transitive, point-imprimitive automorphism groups of 2-designs with prime λ, establishing that only three exceptional symmetric designs can occur and otherwise the group is affine or almost simple, yielding a point-primitive action. The approach builds a double-quotient framework with quotient designs D0 and D1 via a minimal partition, then applies a sequence of combinatorial reductions and group-theoretic classifications (Dembowski, Devillers–Praeger, Camina–Zieschang, Aschbacher) to constrain parameter families. A comprehensive case split—affine vs projective vs elations, almost simple, and various T-types (sporadic, alternating, exceptional Lie, and classical)—ensures that all possibilities are exhaustively tested and either allowed as explicit exceptions or ruled out. The results sharpen understanding of symmetry in combinatorial designs and connect to deep group-theoretic classifications, with implications for translation planes and related incidence geometries. The paper also pursues the open design existence questions associated with the Fermat-prime cases alluded to in the abstract.
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 $λ=1$ acts point-primitively on $\mathcal{D}$. In this paper, we show that the Higman and McLaughlin theorem is still true when $λ$ 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.