Flag-transitive point-primitive automorphism groups of non-symmetric 2-(v, k, 3) designs

Reduction for flag‐transitive point‐primitive 2‐(v, k, λ) designs with λ prime

“…Proposition 2.2 (Liang and Zhou [9,10]). Let  be a nontrivial nonsymmetric 2-v k λ ( , , ) design with ∈ λ {2, 3}, and G be a primitive flag-transitive automorphism group of .…”

“…Suppose that T is a simple group of Lie type over the field q . Then ≥ p T q ( ) except for T = PSL (9) 2 with p T ( ) = 6. In particular, p T ( ) =…”

“…The orders of T Out( ) are listed in [8, Table 5.1.A] for simple classical groups and [8, Table 5.1.B] for simple exceptional groups. Clearly, 9 (by ( 6)). Since T is a transitive group of degree ω, Lemma 2.3 gives…”

“…Since b and z are integers, Equation (9) implies that there exists an integer a such that zω z a λ z z ( + + 2) = 8 ( + 2) ( + 1). .…”

Reduction for flag‐transitive 2‐(v,k,λ) designs with (r,λ)=2

“…A natural problem is whether every nonsymmetric design satisfies the reduction theorem? In fact, Liang and Zhou in [7,8] obtained the same result, that is to say, if and is nonsymmetric, then must be of affine or almost simple type. Furthermore, it is proved in [7] that there are exactly two nontrivial nonsymmetric 2‐ designs admitting a flag‐transitive point‐primitive automorphism group with alternating socle (): one is a 2‐ design with , another is a 2‐ design with , or .…”