Low Dimensional Classical Groups And Flag-Transitive Point-Primitive2-(v, K,4) Symmetric Designs

Groups and combinatorial designs are one of the most important branches of alge-braic combinatorics. There exists deep relationship between groups and designs. We canbetter understand the structures of some groups by studying for the automorphism groupsof designs, and also find some new designs. In1986, A. Delandtsheer studied2-(v, k,1)designs which have some low dimensional classical groups as their automorphism groups.Since then there are many researches on2-(v, k, λ) designs with block-transitive or flag-transitive automorphism groups, such as λ=2or3. In2010, Professor Shenglin Zhouthe case for λ=4and G is2-dimensional classical groups P SL(2, q) as flag-transitivepoint-primitive automorphism group of the symmetric designs. In this paper, we studyflag-transitive point-primitive2-(v, k,4) symmetric designs with some low dimensionalclassical groups as their automorphism groups.Chapter1, we introduce some research advancement on groups and designs. Chapter2, we introduce some basic knowledge on group theory and design theory, especially someelementary properties on2-(v, k,4) symmetric designs. Chapter3, which is main part ofthe paper, we obtain the following two main results.Theorem3.1If D is a non-trivial2-(v, k,4) symmetric design admitting a flag-transitive, point-primitive automorphism group G. Then G can not the groups P SU(3, q),P Sp(4, q)(q=pn, p>2, p is prime) or the socle of G can not be P SL(3, q)(q>2).Theorem3.2If D is a non-trivial2-(v, k,4) symmetric design admitting a flag-transitive, point-primitive automorphism group G=P SL(4, q)(q=2n). Then D is a2-(15,8,4) symmetric design.