Discrete and Combinatorial Geometry of finite Groups

关联方案是一种离散组合几何。设X是集合X上的传递置换群，则在X×X上G的轨道定义了一个关联方案。在本研究中，我们研究了关联方案。我们与a . hanaki合作，对高达28阶的关联方案的同构类进行了分类。我们用电脑。为了构造关联方案，我们使用了C语言编写的程序；为了计算关联方案之间的同构，我们使用了gap语言编写的程序。大多数得到的关联方案可以说是由群给出的。也有一些例外，但几乎所有的都有一个小的秩，对应于在群情况下X×X上轨道的数目，它们可以说包含在少数种类中。正则群是作为关联方案的置换表示。他们被称为瘦。有一个类叫做准瘦。我们的分类找到了一个不属于准瘦类的群给出的例子。这似乎是对未来研究的一个暗示。研究了该程序计算同构的一个应用。如果一个关联模式是由一个组定义的，那么对其自身的同构包含该组的规范化。有几个程度相当小的传递群，它们的归一化器很难计算。应用程序减少了回溯算法在归一化器计算中的搜索空间，实现了数秒内的归一化器计算。我们在程序中使用了代数技术，它对X×X上有很多轨道的群特别有效。我们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.