Some Results On Association Quai-Thin Scheme

In this thesis,we mainly study the relationship between the number of the or-thogonals and schurity and separability of association schemes.We also study the re?lationship between the thin residue and the thin radical of a Klein quasi-thin scheme.First of all,by introducing the concept and properties of the orthogonal,we get that a quasi-thin scheme ? is a Klein scheme if and only if the set S1 consists of either two orthogonals or three thin orthogonals which generate the Klein group.By us-ing the properties of the point extension of the quasi-thin scheme,we conclude that when the order of the Klein quasi-thin scheme X is not less than 9 and the number of the orthogonals is more than 2,? is schurian and separable.Finally,by using the properties of thin residue,we obtain if the number of the orthogonals of a quasi-thin scheme X is at most one,the scheme X is schurian and separable.The following statement is the main result of the thesis:·Let ? be a scheme,and T???S???.If S₁T,then S_?/S₁=T_?/S₁.Moreover,when |T|=2 and T is closed,then X is schurian and separable.In particular,if ? is a quasi-thin scheme and the number of orthogonals is not equal to 2 or 3,then ? is schurian and separable.On the other hand,the thin residue and thin radical of a Klein quasi-thin scheme are not equal in general.In this thesis,we construct an example which is a Klein quasi-thins scheme such that the thin radical of it is equal to the thin residue.·Let M is an even order cyclic group,and the group G is direct product of M group of order 4.Let a be a generator of M,and b a element of order 2 and A4,the alternating of A4.Set H=<a,b>,and ?={Hx:x?G},K is the permutation group induced by right multiplication action of group G on the set ?.Thus X=??,S?=Inv?K,??is a Klein quasi-thin scheme of index 3.In particular,the thin residue is the same as the thin radical.