Block-transitive 2-(v,k,1)2-(v,k,1) designs and the Chevalley groups F4(q)F4(q)

This paper is a contribution to the study of the automorphism groups of 2-(v,k,1)2-(v,k,1) designs. Our aim is to classify pairs (D,G)(D,G) in which DD is a 2-(v,k,1)2-(v,k,1) design and G   is a block-transitive group of automorphisms of DD. It is clear that if one wishes to study the structure of a finite group acting on a 2-(v,k,1)2-(v,k,1) design then describing the socle is an important first step. Let G   act as a block-transitive and point-primitive automorphism group of a 2-(v,k,1)2-(v,k,1) design DD. Set k2=(k,v-1)k2=(k,v-1). In this paper we prove that when q=paq=pa for some prime power and q   is “large”, specifically, q⩾22(k2k-k2+1)a, then the socle of G   is not F4(q)F4(q).