The existence and construction of a family of block-transitive 2- designs

Abstract: Let G be a block-transitive automorphism group of a 2- design . It has been shown that the pairs fall into three classes: those where G is unsolvable and is flag-transitive, those where G is a subgroup of , and those where G is solvable and is of small order. Not much is known about the latter two classes. In this paper, we investigate the existence of 2- designs admitting a block-transitive automorphism group . Using Weil''s theorem on character sums, the following theorem is proved: if a prime power q is large enough and then there is a 2- design which has a block-transitive, but nonflag-transitive automorphism group G. Moreover, using computers, some concrete examples are given when q is small. [Copyright &y& Elsevier]