Primitive Permutation Groups And Flag-transitive Symmetric Designs

After the classifcation of fnite simple groups was completed, the research on somecombinatorial structures has also made great advancement. Combinatorial design is one ofthem. There is a deep inner relationship between the theory of combinatorial designs withsome kinds of symmetry properties and the theory of fnite groups, and these symmetryproperties are mainly refected by all kinds of transitivities of automorphism groups ofdesigns. The research on the symmetry properties of combinatorial designs can help usfnd new designs and classify designs. Conversely, studying the symmetry properties ofdesigns can make us better understand the structure of some groups.A symmetric design is a block design with equal numbers of points and blocks. Nowour attention is focused on the symmetric2-（v, k, λ） designs with λ small, and theirautomorphism groups have some transitivities. In this paper, we study the classifcationof fag-transitive symmetric designs admitting point-primitive automorphism groups withλ≤10. The main results are the following.Theorem2.1.1. Let D be a nontrivial symmetric2-（v, k,4） design, G≤Aut（D） befag-transitive, point-primitive and Soc（G） be an alternating group Anfor n≥5. Then（v, k）=（15,8） and D=（P, B） is one of the following.（1） The points of D are those of the projective space P G（3,2） and the blocks arethe complements of the planes of P G（3,2）, G=A₇or A₈, and the stabilizer Gxof apoint x of D is L₃（2） or AGL3（2） respectively.（2） The points of D are the edges of the complete graph K6and the blocks arethe complete bipartite subgraphs K2,4of K6, G=A6or S₆, and Gx=S4or S4×Z2respectively.Theorem3.1.1. Let D=（P, B） be a nontrivial symmetric2-（v, k, λ） design with5≤λ≤10, G≤Aut（D） be fag-transitive, point-primitive and Soc（G）=Anfor n≥5.Then D is the unique symmetric2-（35,18,9） design which is the complement of thesymmetric2-（35,17,8） design E=（P, B^′）, where the points of E are the lines in a4-dimensional space over GF （2） and blocks have form b（x）={x}∪{y∈P|x∩y=0},where x∈P. Further, G=A₇, S₇, A₈or S₈, and the stabilizer G_x=（S₄×S₃）∩A₇,S₄×S₃, V16·（S₃×S₃） or S₄S2respectively.Theorem4.1.1. Let D be a nontrivial symmetric2-（v, k,3） design（triplane） andlet G≤Aut（D） be fag-transitive and point-primitive. If G is an afne group, then G≤AΓL1（q）, where q is some power of a prime p and p≥5.