Flag-transitive point-primitive non-symmetric $2$-$(v,k,2)$ designs with alternating socle

Abstract

This paper studies flag-transitive point-primitive non-symmetric 2-(v, k, 2) designs. We prove that if D is a non-trivial non-symmetric 2-(v, k, 2) design admitting a flagtransitive point-primitive automorphism group G with Soc(G) = A n for n ≥ 5, then D is a 2-(6, 3, 2) or 2-(10, 4, 2) design.MR (2000) Subject Classification: 05B05, 05B25, 20B25

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

“…For any , by the basic equation and inequality, the above facts force . By [9, Lemma 2.6], we get . If , then , moreover, the orbits of are 1 and .…”

“…Thus We refer to the definition of ‐cyclic partitions in [3], the set of ‐cyclic partitions with respect to (a partition of into classes each of size ) is the union of orbits of on for . By [9, Proposition 3.3] and Lemma 2.2(3), we get . Assume that , then and , that is, , note that …”

On flag-transitive 2-(v,k,2) designs

“…If G is point-primitive, then by [15] and [24], G is of affine or almost simple type. Thus we may assume that G leaves invariant a non-trivial partition (16,6) then by Lemma 2.1, it follows that D is symmetric and hence, in the light of the discussion before the statement of Theorem 1.1, in this case Theorem 1.1(i) holds. Hence we may assume further that (v, k) = (16,6).…”

“…Since then, there have been efforts to classify 2-(v, k, 2) designs D admitting a flag-transitive group G of automorphisms. Through a series of papers [24,25,26,27], Regueiro proved that, if D is symmetric, then either (v, k) ∈ { (7,4), (11,5), (16,6)}, or G AΓL(1, q) for some odd prime power q. Recently, Zhou and the second author [15] proved that, if D is not symmetric and G is point-primitive, then G is affine or almost simple.…”

“…Regueiro [24,25,26,27] has classified all such examples where the design is symmetric (up to those admitting a onedimensional affine group). In the non-symmetric case, the second author and Zhou have dealt with the cases where the socle Soc(G) is a sporadic simple group or an alternating group, identifying three possibilities: namely (v, k) = (176, 8) with G = HS, the Higman-Sims group in [15], and (v, k) = (6,3) or (10,4) with Soc(G) = A v in [16]. Our contribution is the case where Soc(G) = PSL(n, q) for some n 3 and q a prime power.…”