On the automorphisms of 2−(v,k,1)2−(v,k,1) designs

Thirty years ago, a six-person team classified the pairs (D,G)(D,G) where DD is a 2−(v,k,1)2−(v,k,1) design and G   is a flag-transitive automorphism group of D,D, with the exception of those in which G   is a one-dimensional affine group. Since then the effort has been to classify those D,D, designs which are block-transitive but not flag-transitive. This paper contributes to the program for determining the pairs 2−(v,k,1)2−(v,k,1) in which (D,G)(D,G) has a block-transitive group G   of automorphisms. It is clear that if one wishes to study the structure of a finite group acting on a DD design then describing the socle is an important first step. Here we prove that if G   is a block-transitive group of automorphisms of 2−(v,k,1)2−(v,k,1) which has DD as its socle then T   is also transitive on the blocks of T=2F4(q)T=2F4(q).