Classification of 2-transitive symmetric designs

All symmetric designs are determined for which the automorphism group is 2-transitive on the set of points. This note contains a proof of the following result. Theorem. Let D be a symmetric design with v > 2k such that Aut D is 2-transitive on points. Then D is one o f the following: (i) a projective space; (ii) the unique Hadamard design with v = I 1 and k = 5; (iii) a unique design with v = 176, k = 50 and 2 = 14; or (iv) a design with v = 2 2=, k = 2"-*(2 = -1) and 2 = 2"t (2 =-1 1), of which there is exactly one for eacti m >_ 2. The designs in (iv) are discussed in detail in [3 3 . The theorem will be proved as a simple consequence of the classification of finite simple groups. The proof is easier that that of the analogous result [5] for designs with 2 = 1. These two papers clarify the extent to which I-4] is now obsolete. Proof. Let G be a subgroup of Aut D that is 2-transitive on points. Then G is also 2transitive on blocks, and these two 2-transitive permutation representations are inequivalent; in particular, the stabilizer Gx of a point x is not conjugate to the stabilizer GB of a block B. Note that we may replace G by any 2-transitive subgroup of G. A list of 2-transitive groups is contained in ES]; compare [1]. We only need to check whether a group on the list has two inequivalent 2-transitive permutation representations of the same degree (and having the same permutation character). When G has a nonabelian simple normal subgroup, this is, in effect, already contained in [1], and leads to (i)-(iii). Assume that G does not have a nonabelian simple normal subgroup. Then G <_ AGL(d,p) for some prime p, and G contain the translation group V. We can iden{ify V with the set of points of D, and then let x = 0. Now G = VG o = VG a, so that Go and Gn are nonconjugate complements to V in G. If Z(Go) # 1 then Go = N~(Z(Go)) is conjugate to G8 = N~(Z(Gs)) (since Z(Go) * This research was supported in part by NSF Grant MCS 7903130-82.