Applications Of Finite Groups In Block Designs

It is an important project to classify groups and combinatorial designs where the group acts on the blocks of the design. The construction of 2-designs with certain transitive prop-erties, for example, flag or block-transitivity, is a hot topic with great theoretic significance and application value in the field of combinatorial design theory.In 1990, Buekenhout, Delandsheer, Deoyen, Kleidman, Liebeck and Saxl classified the pairs (S, G) where S is a finite linear space which has a flag-transitive group G of automor-phisms, with the exception of those in which G is a one-dimensional affine group. Since then the effort has been to classify those linear spaces which are line-transitive but not flag-transitive. This thesis contributes to this program. The thesis is divided into five chapters, we mainly study line-transitive linear spaces, existence of block-transitive 2-(v, k,1) designs and the classification of block-transitive 2-(v,17,1) designs.The first chapter is devoted to surveying research background results, methods in this area.In the second chapter, we collect some notations and definitions, and quote some pre-liminary results which will be used in this thesis. We give those related to abstract group theory, permutation groups, combinatorial designs and linear spaces theory in separate sec-tions.In the third chapter, we study the automorphisms group of line-transitive linear spaces, and we discuss the reduction problem of almost simple groups G which act as line-transitive automorphism group of finite linear spaces with socles T-Sz(q),2F4(q), F4(q). We will show to derive the case where the socle T acts line-transitively on S from the case where G acts line-transitively on S, which provides an important basis for determining all these finite linear spaces.In the fourth chapter, we investigate the existence of block-transitive 2-(v, k,1) designs admitting almost simple groups as their automorphism groups whose socles are Lie simple L(q). We find an appropriate q-bound for these designs, which help to narrowed down the search of the block-transitive 2-(v, k,1) designs.In the five chapter, we study the block-transitive 2-(v,17,1) designs, and make use of Delandtsheer-Doyen theory, primitive factor、theory about fixed point and parameter relation in designs to discuss the properties and structure of automorphism groups of these type of designs, and also their classification.