SGDs with doubly transitive automorphism group

Symmetric graph designs, or SGDs, were deened by Gronau et al. as a common generalisation of symmetric BIBDs and orthogonal double covers. This note gives a classiication of SGDs admitting a 2-transitive automorphism group. There are too many for a complete determination, but in some special cases the determination can be completed, such as those which admit a 3-transitive group, and those with = 1. The latter case includes the determination of all near 1-factorisations of Kn (partitions of the edge set into subsets each of which consists of disjoint edges covering all but one point) which admit 2-transitive groups. (c) any edge of K n lies in exactly of the graphs X 1 ; : : :; X n. To avoid degenerate cases, we assume that X is neither the complete nor the null graph on n vertices. Symmetric graph designs generalise both symmetric BIBDs (symmetric 2-designs, the case where X = K k , F = K), and orthogonal double covers or ODCs (the case where = 2 and F is a single edge). I refer to Gronau et al. 3] or Rosa 4] for elementary properties of SGDs. In particular, X has (n ? 1)=2 edges; F has (? 1)=2 edges. In this note I give a classiication of SGDs admitting a doubly transitive automorphism group. There are too many of these designs for a complete list to be possible, so what follows is a structural description. By the complement of a SGD, I mean the structure obtained by replacing each X i by its complement in K n. I do not know whether this always gives another SGD, but it does so in the 2-transitive case. The results of this note, except for Corollary 1, use the Classiication of Finite Simple Groups.