Automorphism groups of designs

Abstract

From a geometric point of view, the most interesting designs (see w 2 for definitions) are generally those admitting fairly large automorphism groups. The methods of finite permutation groups may be applied to such designs, and vice versa, as in [5, 6, 8, 11, 13 and 143. We shall prove several general results which are useful in the study of automorphism groups of designs, and then use some of these to characterize some designs admitting large automorphism groups. Further applications are found in [11]. A Hadamard design is a symmetric design with k = (v- 1)/2 (see [3] or [17] for the connection with Hadamard determinants). The best known examples of such designs - other than the Desarguesian projective spaces over GF(2) - are the Paley designs ([15]; cf. [183 and [11]). The points of a Paley design are the elements ofF = GF(v), where v > 3 is a prime power -=- 3 (rood 4), while the blocks are the translates under F + of the set Q of non-zero squares of F. This design admits an automorphism group of odd order {x --+ x ~ t + a[t E Q, a ~ F, ~Aut(F)} which is transitive on incident point-block pairs; this group is not always the full automorphism group (cf. [11]). Theorem 1.1. Paley designs are the only Hadamard designs admitting automorphism groups which are transitive on incident point-block pairs but which are not 2-transitive on points, A larger class of designs will also be considered, all related to F. Our characterizations of some of these designs generalize many of the results of Liineburg [13], whose approach is different.