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Almost simple groups as flag-transitive automorphism groups of 2-designs with {\lambda} = 2
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abstract
In this article, we study $2$-designs with $\lambda=2$ 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 $\lambda=2$ 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 $(176,8,2)$.
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Cited by 2 Pith papers
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Flag-transitive point-primitive quasi-symmetric $2$-designs and exceptional groups of Lie type
For quasi-symmetric 2-designs with intersection numbers 0 and y between 2 and 10, the socle of any flag-transitive point-primitive automorphism group cannot be an exceptional group of Lie type.
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On flag-transitive automorphism groups of $2$-designs with $\lambda$ prime
For flag-transitive point-primitive 2-designs with λ prime, exceptional simple socles give exactly the Suzuki-Tits ovoid design and a new G2(q) coset-geometry family, while sporadic socles give only three known designs.
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