The automorphism group of a flag-transitive 6-(v, k, 2) design is a 3-homogeneous permutation group. Therefore, using the classification theorem of 3-homogeneous permutation groups, the classification of flag-transitive 6-(v, k,2) designs can be discussed. In this paper, by analyzing the combination quantity relation of 6-(v, k, 2) design and the characteristics of 3-homogeneous permutation groups, it is proved that: there are no 6-(v, k, 2) designs D admitting a flag transitive group G ≤ Aut (D) of automorphisms.

In this article, we show that if D is a nontrivial nonsymmetric 2 − (v, k, 2) design admitting a flag-transitive point-primitive automorphism group G, then G must be an affine or almost simple group. Moreover, if the socle of G is sporadic, then D is the unique 2 − (176, 8, 2) design with G = H S, the Higman-Sims simple group. , a m )π | a i ∈ Aut(T ), π ∈ S m , a i ≡ a j (mod Inn(T )) for all i, j }, and there is an α ∈ P such that G α ≤ {(a, . . . , a)π | a ∈ Aut(T ), π ∈ S m } ∼ = Aut(T ) × S m , and M α = D = {(a, a, . . . , a) | a ∈ I nn(T )} Journal of Combinatorial Designs

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

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This paper is devoted to the classification of flag-transitive 2-(v, k, 2) designs. We show that apart from two known symmetric 2-(16, 6, 2) designs, every flag-transitive subgroup G of the automorphism group of a nontrivial 2-(v, k, 2) design is primitive of affine or almost simple type. Moreover, we classify the 2-(v, k, 2) designs admitting a flag transitive almost simple group G with socle PSL(n, q) for some n 3. Alongside this analysis we give a construction for a flag-transitive 2-(v, k − 1, k − 2) design from a given flag-transitive 2-(v, k, 1) design which induces a 2-transitive action on a line. Taking the design of points and lines of the projective space PG(n − 1, 3) as input to this construction yields a G-flag-transitive 2-(v, 3, 2) design where G has socle PSL(n, 3) and v = (3 n − 1)/2. Apart from these designs, our PSL-classification yields exactly one other example, namely the complement of the Fano plane.