Almost simple groups as flag-transitive automorphism groups of 2-designs with λ = 2
Seyed Hassan Alavi
TL;DR
The paper addresses the existence and classification of $2$-designs with $\lambda=2$ that admit a flag-transitive almost simple automorphism group whose socle is an exceptional group of Lie type, proving nonexistence in this case. It employs a strategy combining large-subgroup bounds $|H|^3\ge |G|$ and the ABD-Exp classification to exhaust possibilities for point-stabilizers, ultimately ruling out all exceptional-socle scenarios and aligning the results with existing classifications. The main contribution is a complete taxonomy of nontrivial $2$-designs with $\lambda=2$ under flag-transitive, point-primitive almost simple groups: either an infinite family with parameters $((3^n-1)/2,3,2)$ and $X=\mathrm{PSL}_n(3)$ for $n\ge 3$, or one of the listed finite designs (e.g., $(6,3,2)$, $(7,4,2)$, $(10,4,2)$, $(11,5,2)$, $(28,7,2)$, $(28,3,2)$, $(36,6,2)$, $(126,6,2)$, $(176,8,2)$). This sharpens the understanding of how exceptional groups of Lie type can act on combinatorial designs and guides future classification efforts in incidence geometry with restricted automorphism groups.
Abstract
In this article, we study $2$-designs with $λ=2$ admitting a flag-transitive almost simple automorphism group with socle a finite simple exceptional group of Lie type, and we prove that such a $2$-design does not exist. In conclusion, we present a classification of $2$-designs with $λ=2$ admitting flag-transitive and point-primitive automorphism groups of almost simple type, which states that such a $2$-design belongs to an infinite family of $2$-designs with parameter set $((3^n-1)/2,3,2)$ and $X=PSL_n(3)$ for some $n\geq 3$, or it is isomorphic to the $2$-design with parameter set $(6,3,2)$, $(7,4,2)$, $(10,4,2)$, $(10,4,2)$, $(11,5,2)$, $(28,7,2)$, $(28,3,2)$, $(36,6,2)$, $(126,6,2)$ or $(176,8,2)$.