Imprimitive Geometries and Groups

Cayley的一个结果表明，每个群都有一个忠实的表示为传递置换群。许多都有原始的表示，例如，所有的有限单群。事实上，经典的单群在射影空间或极空间的点上起原始作用。然而，许多群都有非纯正的忠实置换表示，因此必须考虑非原始的忠实表示。在这个项目中，霍尔研究了作为几何的自同构群的有限非本原群。他还考虑了具有自然和非平凡等价关系的几何的逆-描述和刻画。特别有趣的是等价关系，它的类是某个非平凡正规子群的轨道，然后该群自然地嵌入到相应的圈积中。所研究和刻画的几何是高度齐次的部分线性空间，特别是那些与形式的等距群相关的几何，其中群的非本原性可以来自形式的退化。对于次数大于2的形式，这导致了关于双线性形式和相关形式的Witt定理的可能推广的问题。群论通常被描述为对称的数学。也就是说，许多群最好地描述为几何对象的对称性的集合。相反，许多几何图形最好的特征是它们的相关群，这一观点可以追溯到希尔伯特。有限群在数学以外的许多领域都是通过对称性出现的，例如物理、化学和计算机科学。有限对称群的构件是有限单群，并且被实现为本原置换群-在适当意义上有效的置换群。然后，一个一般的有限群由单群粘合在一起，它的对称实现是非本原的-由各种本原成分构成。近年来，在单群及其本原置换表示的刻画方面取得了很大的成功。下一步重要的一步是理解并熟练使用免疫组织。这个项目的目的是给出某些非本原群的初等几何描述，并反过来刻画和识别从非本原对称群的性质开始的几何。

A result of Cayley implies that every group has a faithfulrepresentations as a transitive permutation group. Many have primitivefaithful representations, for instance, all finite simplegroups. Indeed the classical simple groups act primitively on thepoints of a projective or polar space. However many groups have noprimitive faithful permutation representation, so imprimitive faithfulrepresentations must be considered. In this project Hall studiesfinite imprimitive groups that act as automorphism groups ofgeometries. He also considers the converse---description andcharacterization of geometries that come provided with a natural andnontrivial equivalence relation. Of particular interest areequivalence relations whose classes are the orbits of some nontrivialnormal subgroup, the group then naturally embedded in thecorresponding wreath product. The geometries studied andcharacterized are highly homogeneous partial linear spaces,particularly those associated with isometry groups of forms, whereimprimitivity for the group can come from degeneracy of the form. Forforms of degree greater than two, this leads to questions aboutpossible generalizations of Witt's theorem on bilinear and relatedforms.Group theory is often described as the mathematics of symmetry. Thatis, many groups are best described as collections of symmetries ofgeometric objects. Conversely many geometries are best characterizedby their associated groups, a point of view that goes back to Hilbert.Finite groups appear prominently via symmetry in many fields otherthan mathematics, for example, physics, chemistry, and computerscience. The building blocks of finite symmetry groups are the finitesimple groups and are realized as primitive permutationgroups---permutation groups that in an appropriate sense areefficient. A general finite group is then glued together from simplegroups, and its symmetry realization is imprimitive---constructed fromvarious primitive constituents. There has been great success recentlyin the description of simple groups and their primitive permuationrepresentations. Understanding of and facility with imprimitivegroups is the important next step. The project is aimed at givingelementary geometric descriptions of certain imprimitive groups andconversely characterizing and recognizing geometries starting fromproperties of an imprimitive symmetry group.