Discrete and Combinatorial Geometry of finite Groups

结合方案是离散的组合几何。设X是集合X上的传递置换群，则G在X×X上的轨道定义了一个结合方案。在本研究中，我们研究了关联方案。我们与A.Hanaki共同研究了高达28阶的结合方案的同构类。我们用的是电脑。在OEDER中，我们用C语言编写了一个程序来构造结合模式，用GAP语言编写了一个程序来计算结合模式之间的同构。所得到的大部分结合方案可以说是群组给出的。有一些例外，但几乎所有的例外都具有与群情况下X×X上的轨道数相对应的小秩数，可以说它们包含在少数几种类型中。正则群是作为结合方案的置换表示。它们被称为“瘦”。有一类叫做准薄。我们的分类发现了一个不属于准薄类的群所给出的例子。这似乎是对未来研究的一个暗示。我们研究了我们的程序计算同构的一个应用。如果一个结合方案是由一个群定义的，那么它自身的同构包含这个群的正规化子。有几个次数相当小的传递群，其中的正规化子很难计算。在归一化变换中，我们应用我们的程序缩小了回溯算法的搜索空间，并且在几秒钟内就可以计算出这样的归一化。我们在程序中使用了一种代数技巧，它对X×X上有多个轨道的群特别有效。我们现在正在从理论上研究这个程序。

An association scheme is a discrete combinatorial geometry. Let X be a transitive permutation group on a set X.Then the orbits of G on X×X defines an association scheme. In the present research we studied association schemes. We classified the isomorphism classes of association schemes of order up to 28 as a joint work with A.Hanaki, one of the research investigator. We used computers. In oeder to construct association schemes we used a program written by C and for computing isomorphisms between association schemes we used a program written by GAP-language. Most of the obtained association schemes can be said given by groups. There are a number of exceptions but almost all of them have small ranks which correspond to the number of the orbits on X×X in group case, and they can be said to be contained in a small number of kinds. Regular groups are permutation representations as an association scheme. They are called thin. There are a classes called quasi-thin. Our classification found an example not given by a group belonging to quasi-thin class. This seems to be a hint for future research. We studied an application of our program computing isomorphisms. If an association scheme is defined by a group, then isomorphisms to itself contain the normalizer of the group. There are a couple of transitive groups of rather small degree of which normalizers are hard to compute. We applied our program to reduce the searching space of backtrack algorithm in the normalizer comutation and we have been abel to compute such normalizers within several seconds. We used an algebraic technique in the program and it was particularly effective for groups with many orbits on X×X.We are now studying the program theoretically.