Imprimitive Geometries and Groups

Michigan State University, East Lansing MI

Investigators

Abstract

A result of Cayley implies that every group has a faithful representations as a transitive permutation group. Many have primitive faithful representations, for instance, all finite simple groups. Indeed the classical simple groups act primitively on the points of a projective or polar space. However many groups have no primitive faithful permutation representation, so imprimitive faithful representations must be considered. In this project Hall studies finite imprimitive groups that act as automorphism groups of geometries. He also considers the converse---description and characterization of geometries that come provided with a natural and nontrivial equivalence relation. Of particular interest are equivalence relations whose classes are the orbits of some nontrivial normal subgroup, the group then naturally embedded in the corresponding wreath product. The geometries studied and characterized are highly homogeneous partial linear spaces, particularly those associated with isometry groups of forms, where imprimitivity for the group can come from degeneracy of the form. For forms of degree greater than two, this leads to questions about possible generalizations of Witt's theorem on bilinear and related forms. Group theory is often described as the mathematics of symmetry. That is, many groups are best described as collections of symmetries of geometric objects. Conversely many geometries are best characterized by their associated groups, a point of view that goes back to Hilbert. Finite groups appear prominently via symmetry in many fields other than mathematics, for example, physics, chemistry, and computer science. The building blocks of finite symmetry groups are the finite simple groups and are realized as primitive permutation groups---permutation groups that in an appropriate sense are efficient. A general finite group is then glued together from simple groups, and its symmetry realization is imprimitive---constructed from various primitive constituents. There has been great success recently in the description of simple groups and their primitive permuation representations. Understanding of and facility with imprimitive groups is the important next step. The project is aimed at giving elementary geometric descriptions of certain imprimitive groups and conversely characterizing and recognizing geometries starting from properties of an imprimitive symmetry group.