On flag-transitive automorphism groups of 2-designs with λ prime

In this article, we study 2-(v, k, λ) designs D with λ prime admitting flag-transitive and point-primitive almost simple automorphism groups G with socle T a finite exceptional simple group or a sporadic simple group. If the socle of G is a finite exceptional simple group, then we prove that D is isomorphic to one of two infinite families of 2-designs with point-primitive automorphism groups, one is the Suzuki-Tits ovoid design with parameter set (v, b, r, k, λ) = (q² + 1, q²(q² + 1)/(q − 1), q², q, q − 1), where q − 1 is a Mersenne prime, and the other is newly constructed in this paper and has parameter set (v, b, r, k, λ) = (q³(q³ − 1)/2, (q + 1)(q⁶ − 1), (q + 1)(q³ + 1), q³/2, q + 1), where q + 1 a Fermat prime. If T is a sporadic simple group, then we show that D is isomorphic to a unique design admitting a point-primitive automorphism group with parameter set (v, b, r, k, λ) = (176, 1100, 50, 2), (12, 22, 11, 6, 5) or (22, 77, 21, 6, 5).