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Abstract The classification of all block-transitive 2 − (v , k , 1) designs has been known as a long-standing and still open problem. This work is a contribution to the general program to classify the pairs (D , G) in which D is a 2 − (v , k , 1) design and G act as a point-primitive block-transitive automorphism group of D with a Suzuki group Sz (q) socle. We determine the point-stabilizer G α ∩ Sz (q) and the block-stabilizer G B ∩ Sz (q), and the parameter of D is given. [ABSTRACT FROM AUTHOR]

Thirty years ago, a six-person team classified the pairs ( D , G ) where D is a 2 − ( v , k , 1 ) design and G is a flag-transitive automorphism group of 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 , designs which are block-transitive but not flag-transitive. This paper contributes to the program for determining the pairs 2 − ( v , k , 1 ) in which ( 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 D 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 ) which has D as its socle then T is also transitive on the blocks of T = 2 F 4 ( q ) . [ABSTRACT FROM AUTHOR]