Discrete Mathematics 57 (1985) 1-7 North-Holland AUTOMORPHISM GROUPS OF BLOCK DESIGNS WHICH ARE BLOCK TRANSITIVE A.R. CAMINA School of Mathematics and Physics, University of East Anglia, Norwich, NR4 7TJ, United Kingdom Received 15 May 1984 1. Introduction In this introduction we discuss some of the results which are thought to be of most interest which are going to be proved in the later sections. The main purpose of the article is to investigate some of the implications on the structure of a 2-(v, k, h) block design that possesses a block-transitive automorphism group. Before stating the first main result it is necessary to introduce a definition which may not be standard. Let X be a permutation group on a set/2. Assume that X has t orbits O1,...,/2, and choose ai ~ Oi, 1 <~i~ < t. Let X~ have t~ orbits on/2. We will define the rank of X to be Y.'~=I t~. If X is transitive this gives the normal definition of rank. However, it is true that if ~r is the permutation character then rank of X is the inner product (Tr, 7r). In that sense this definition is a genuine generalization of the rank of a transitive permutation group. If G is a subgroup of the automorphism group of a 2- (v, k, h) design which is block transitive then the permutation group induced on the points incident with a given block is independent of the block. We can now state our first main theorem. Theorem 1. Let G be a subgroup of the automorphism group of a 2-(v, k, h) design which is transitive on blocks. Let B be a block and suppose that GB induces a permutation group on the points incident with B which has rank m and t orbits. Then G has point rank ~m - t+ 1. A particular corollary is the case where GB acts as the identity group in which case we have that the point rank of G is less than or equal to k 2- k + 1. There is no assumption in this paper that a block is determined by the set of points incident with the block. We should make it clear at this point that an automorphism of a design ~ is a pair (g, h) where g is a permutation on the points of ~ and h is a permutation on the blocks of ~ such that ot/B if and only if ctg/Bh for all points a and all blocks B. 0012-365X/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)