Almost simple groups as flag-transitive automorphism groups of 2-designs with {\lambda} = 2

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 $(176,8,2)$

On flag-transitive 2-(k^2, k, λ) designs with λ | k

It is shown that, apart from the smallest Ree group, a flag-transitive automorphism group G of a 2-(k^2, k, λ) design D, with λ | k, is either an affine group or an almost simple classical group. Moreover, when G is the smallest Ree group, D is isomorphic either to the 2-(6^2, 6, 2) design or to one of the three 2- (6^2, 6, 6) designs constructed in this paper. All the four 2-designs have the 36 secants of a non-degenerate conic C of PG(2,8) as a point set and 6-sets of secants in a remarkable configuration as a block set