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

Abstract

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.

On flag‐transitive imprimitive 2‐designs

“…Lemma 5(vi) yields | k c c 16 ( − 1) which is satisfied in each remaining case. Combining Lemma 5(iii) and (v) yields d c = By Lemma 5(iii) r c = 4( − 1).Thus the possibilities for ℓ c d k r ( , , , , ) are(6,16,20,20,2),(10, 28, 32, 36,2), (26, 76, 80, 100, 2). (iv) ℓ x ( , ) = (3, 1).…”

“…For λ = 2 the bound in Proposition 7 gives ⩽ k 24. Together with Liang and Xia, the authors showed in [10] that there are only two imprimitive flag-transitive 2-designs, both of which are 2 − (16, 6, 2) designs. Thus this bound is definitely not tight for all λ.…”

“…Let all parameters be as in Lemma 5. If λ = 2, we showed in [10] (by group-theoretic arguments) that k…”

“…Recall the result of Higman and McLaughlin [11] that for λ = 1, all flag-transitive 2-designs are point-primitive. A recent result of the authors and colleagues in [10,Theorem 1.1] shows that there are up to isomorphism just two designs which prevent this conclusion holding also for λ = 2: namely there are exactly two flag-transitive, pointimprimitive 2-v k ( , , 2) designs, and both of them are 2 − (16, 6, 2) designs. For the cases λ = 3, 4, we list in Proposition 8 all "numerically feasible" parameter sets for flag-transitive, point-imprimitive 2-designs, that is to say, parameter sets which satisfy all the conditions imposed by our preliminary results in Section 2.…”

Correction to: Flag-transitive block designs and unitary groups

“…Therefore, we correct Theorem 1.1 in [2] as below: Theorem 1.1 Let D be a nontrivial 2-design with gcd(r , λ) = 1, and let α be a point of D. Suppose that G is an automorphism group of D whose socle is X = PSU(n, q) with (n, q) = (3, 2). If G is flag-transitive, then λ ∈ {1, 2, 3, 5} and v, k, λ, X α and X are as in one of the lines in Table 1 or one of the following holds: It is worth noting by [6] that there is a general construction method for 2-designs from linear space:…”