Imprimitive Geometries and Groups

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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.