Discrete and Combinatorial Geometry of finite Groups

结合方案是一个离散的组合几何。设X是集合X上的可迁置换群，则G在X×X上的轨道定义了一个结合概型.在本研究中，我们研究了关联方案。我们与研究者之一的A.Hanaki合作，对阶数不超过28的结合方案的同构类进行了分类。我们用电脑。为了构造关联方案，我们使用C语言编写的程序，为了计算关联方案之间的同构，我们使用GAP语言编写的程序。大多数得到的结合方案可以说是由群给出的。有一些例外，但几乎所有的例外都有小的秩，对应于群情况下X×X上轨道的数目，可以说它们包含在少数种类中。正则群是作为关联方案的置换表示。他们被称为薄。有一类叫准瘦。我们的分类发现了一个不属于准薄类的组给出的例子。这似乎是对未来研究的一个提示。我们研究了我们的程序计算同构的应用。如果一个结合模式是由一个群定义的，那么它自身的同构包含这个群的正规化子。有几个传递群的度相当小，其正规化子很难计算。我们应用我们的程序来减少回溯算法在正规化子计算中的搜索空间，我们已经能够在几秒钟内计算出这样的正规化子。我们在程序中使用了一种代数技巧，它对X×X上有许多轨道的群特别有效。我们现在正在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.