The - Designs with Unsolvable Block Transitive Automorphism

This paper is a contribution to the study of the automorphism groups of - designs. Let be - design and Aut a block transitive and a point primitive. If is unsolvable, then Soc , the socle of , is not . 1. Introduction A design is a pair consisting of a finite set of points and a collection of of , called blocks, such that any 2-subsets of are contained in exactly one block. We will always assume that . Let be a group of automorphisms of a design . Then is said to be block transitive on if is transitive on and is said to be point transitive (point primitive) on if is transitive (primitive) on . A flag of is a pair consisting of a point and a block through that point. Then is flag transitive on if is transitive on the set of flags. The classification of block transitive designs was completed about thirty years ago (see [1]). In [2], Camina and Siemons classified designs with a block transitive, solvable group of automorphisms. Li classified designs admitting a block transitive, unsolvable group of automorphisms (see [3]). Tong and Li [4] classified designs with a block transitive, solvable group of automorphisms. Han and Li [5] classified designs with a block transitive, unsolvable group of automorphisms. Liu [6] classified (where ) designs with a block transitive, solvable group of automorphisms. In [7], Han and Ma classified designs with a block transitive classical simple group of automorphisms. This paper is a contribution to the study of the automorphism groups of designs. Let be design and a block transitive and a point primitive. We prove the following theorem. Main Theorem. Let be design and a block transitive and a point primitive. If is unsolvable, then . 2. Preliminary Results Let be a design defined on the point set and suppose that is an automorphism group of that acts transitively on blocks. For a design, as usual, denotes the number of blocks and denotes the number of blocks through a given point. If is a block, denotes the setwise stabilizer of in and is the pointwise stabilizer of in . Also, denotes the permutation group induced by the action of on the points of , and so . Lemma 1 (see [8]). Let , , , then every maximal subgroup of is conjugate to one of the following: (1) ; (2) ; (3) ; (4) ; (5) , if ; (6) ; (7) ; (8) ; (9) ; (10) ; (11) ; (12) , where and be prime. Lemma 2 (see [9]). Let be an exceptional simple group of Lie type over , and let be a group with . Suppose that is a maximal subgroup of not containing , then one of the following holds: (1) ; (2) , or , if ; (3) is a parabolic subgroup of . Lemma 3 (see [7]). Let and be a