Block-transitive, point-imprimitive designs with lambda = 1

Abstract Let D be a 2-( v , k ,1) design with a group G of automorphisms which is transitive on the blocks of D and transitive but imprimitive on the points of D . Delandtsheer and Doyen (1989) proved that v is bounded above by (k−2) 2 (k+1) 1 4 . Carrying on from the work of Cameron and Praeger (1989), we show that if v is equal to this upper bound then v =729 and k =8. Further work of Nickel et al. (1992) has shown that, up to an isomorphism, there are 467 block-transitive, point-imprimitive 2-(729,8,1) designs.