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

This paper is a contribution to the study of the automorphism groups of 2 - ( v , k , 1 ) designs. Our aim is to classify pairs ( D , G ) in which D is a 2 - ( v , k , 1 ) design and G is a block-transitive group of automorphisms of D . It is clear that if one wishes to study the structure of a finite group acting on a 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 ) design D . Set k 2 = ( k , v - 1 ) . In this paper we prove that when q = p a for some prime power and q is "large", specifically, q � 2 2 ( k 2 k - k 2 + 1 ) a , then the socle of G is not F 4 ( q ) .