Block Transitive Automorphism Groups Of 2-(v, K, 1) Designs

This thesis is devoted to classifying block designs whose automorphism group G acts transitively on blocks. It consists of four chapters.In Chapter 1, we introduce the background of groups and designs and summary the main results.In Chapter 2, we introduce some basic contents of groups and block designs.The classification of block-transitive 2 - (v,3,1) designs was completed more than twenty years ago. In [10] Camina and Siemens classified 2 - (v. 4,1) designs with a block-transitive, solvable groups of automorphisms. Li classified 2- (v, 4,1) designs admitting a block-transitive, unsolvable automorphism groups (see [49]). In [50] Li and Tong classified 2 - (v, 5.1) designs with a block-transitive, solvable groups of automorphisms. So for block transitive 2 - (v, 5.1) designs, we need to study the case in which the given block transitive group of automorphisms is unsolvable. In Chapter 3 we consider this case and prove the following theoremTheorem 3.1 If a block-transitive group of automorphisms of a 2 - (v. 5,1) design is unsolvable, then it is flag-transitive.By Theorem 3.1 and the result of Li and Tong [50], the classification of block-transitive 2 - (v, 5,1) designs was completed. And we haveCorollary 3.2 Let D be a 2 - (v, k, 1) design, G Aut(D) be block-transitive but not flag-transitive, then G is solvable and one of the following is true(1) if G is point imprimitive, then v - 21, and G Z21 : Z6;(2) if G is point primitive, then G A L(1,v) and v = pa. where p is a prime number with p = 21 (mod 40), and a an odd integer.In Chapter 4, we consider 2 - (v, k. 1)(k = 6, 7, 8, 9) designs with a block-transitive, unsolvable groups of automorphisms. We get the following resultTheorem 4.1 Let D be a 2 - (v, k, 1) (k = 6, 7, 8, 9)design with G Aut(D) block-transitive, point-primitive but not flag-transitive. If G is unsolvable. then we have(1) if k = 6 and (k,v) 2, then the socle of G is not the exceptional simple groups of Lie type.(2) if k = 7, then the socle of G is not the exceptional simple groups of Lie type.(3) if k = 8, then the socle of G is not the exceptional simple groups of Lie type.(4) if k = 9, then but two groups (E6(3), 2E6(3)), the socle of G is not the exceptional simple groups of Lie type.