Mathieu Groups And Flag-transitive 2-(v,k,λ)Designs

There are important internal connections between the group theory and the theory of combinatorial designs,which reflected by the flag-transitivity,point-primitivity and other symmetries of the automorphism groups.We can find more new designs and obtain the classification of some designs by researching automorphism groups of the design,for they are interacted and contributed on each other.Flag-transitivity is one of important conditions that can be imposed on the automorphism group of a design.Dembowski has proved in Finite Geometries that if G ≤ Aut(D)is flag-transitive and(v-1,k-1)≤2,then G is also point-primitive.In 1987,Davies proved that if the socle of G is a sporadic simple group then G does not act flag-transitively on a 2-(v,k,1)design.In 1990,Buekenhout,Delandtsheer,Doyen,Kleidman,Liebeck,Saxl classified flag-transitive finite linear spaces(except one-dimensional affine type).In 1998,P.H.Ziechang has proved that if G<Aut(D)is flag-transitive and(r,λ)= 1,then G is of affine type or almost simple type.In 2013,Tian and Zhou completed the classification of symmetric 2-(v,k,λ)designs admitting a flag-transitive,point-primitive automorphism group G of almost simple type with sporadic socle.According to this results,we continue to study the classification of designs with(v-1,k-1)≤2 and a flag-transitive automorphism group G.And we obtained the result as follows:Theorem 0.1.If D is a 2-(v,k,λ)designs,with Soc(G)is one of five Mathieu groups Mi and(v-1,k-1)≤2,where i=11,12,22,23 or 24.Then,up to isomirphism,there exists 62 2-designs satisfying the assumption.