Hadamard Tournaments with Transitive Automorphism Groups

Abstract

Let D = (P, B) be a Hadamard 2-( 4A + 3, 2A + 1, A) design, where P and B denote the sets of points and blocks of D, respectively. Let T be a bijection from B to P such that (a) T(a) does not belong to a for every element a of B, and (b) T(a) belongs to f3 if and only if T(f3) does not belong to a for every 2-subset {a, f3} of B. Then we call D, together with a T, a Hadamard tournament. It is known that the Hadamard matrix corresponding to a Hadamard tournament is equivalent to a skew-Hadamard matrix ([7], [8]). Now we are interested in a (D, T) with a transitive automorphism group G, which we simply call a THT. Here an automorphism S of (D, T) is an automorphism of D such that T(a)s = T(as) for any element a of B. Classic and seemingly only examples of THT are quadratic residue (or non-residue) de~igns D(q), where q is a prime power such that q == 3 (mod 4), with appropriate Ts. J. B. Kelly and E. C. Johnsen, independently, characterized D(q) wit}1 q a prime as a THT such that G contains a cyclic transitive subgroup ([4], [5]), and C. W. H. Lam characterized D(q) as a THT such that G is of rank 3 on P ([6]). t We notice that their assumptions imply that 4A + 3 is a prime power. In this paper we establish this fact for larger classes of THT. Two particular such classes are mentioned here: (1) (D, T) such that G contains a regular transitive subgroup. Cases of Kelly, Johnsen and Lam are included in (1). (2) (D, T) such that G contains a supersolvable transitive subgroup. Furthermore, we show that 9p cannot be the order of a THT, where p is a prime larger than 3. We note that G obviously has odd order and hence, by the Feit-Thompson theorem, it is solvable. For basic results on group theory and Hadamard matrices see ([2]) and ([1]), respectively.