The thesis is concerned with group theoretical properties of unitary groups, mainly of II_1 factors. The author gives a new and elementary proof of an result on extreme amenability, defines the bounded normal generation property and invariant automatic continuity property and proves these for various unitary groups of functional analytic types.

The main focus of this thesis is algebraic modules---modules that satisfy a polynomial equation with integer co-efficients in the Green ring---in various finite groups, as well as their general theory. In particular, we ask the question `when are all the simple modules for a finite group G algebraic?' We call this the (p-)SMA property. The first chapter introduces the topic and deals with preliminary results, together with the trivial first results. The second chapter provides the general theory of algebraic modules, with particular attention to the relationship between algebraic modules and the composition factors of a group, and between algebraic modules and the Heller operator and Auslander--Reiten quiver. The third chapter concerns itself with indecomposable modules for dihedral and elementary abelian groups. The study of such groups is both interesting in its own right, and can be applied to studying simple modules for simple groups, such as the sporadic groups in the final chapter. The fourth chapter analyzes the groups PSL(2,q); here we determine, in characteristic 2, which simple modules for PSL(2,q) are algebraic, for any odd q. The fifth chapter generalizes this analysis to many groups of Lie type, although most results here are in defining characteristic only. Notable exceptions include the small Ree groups, which have the 2-SMA property for all q. The sixth and final chapter focuses on the sporadic groups: for most groups we provide results on some simple modules, and some of the groups are completely analyzed in all characteristics. This is normally carried out by restricting to the Sylow p-subgroup. This thesis develops the current state of knowledge concerning algebraic modules for finite groups, and particularly for which simple groups, and for which primes, all simple modules are algebraic.

Formal systems in architectural design aim at the systematic description, interpretation, and evaluation of existing works of architecture as well the systematic creation of new works of architecture. Currently all formal analysis using group theoretical tools focus on repetitive designs that show immediately their recursive structure. It is suggested here that highly complex designs can still be described and analyzed with group theoretical manner. The broader question that is opened up here is whether a complex architecture object or part depending on the interest of the researcher, can be interpreted as a layered object whose parts are all related symmetrically; in other words whether an asymmetric shape or configuration can be understood in terms of nested arrangements of some order of symmetry. The object of analysis has been polemically selected here to be the NY5 architecture, a set of designs that are all clearly exemplifying formal qualities of abstraction, layering, complexity, depth and so on, all appearing impenetrable to a systematic and rigorous analysis using the existing group theoretical formal methods. For example, Richard Meier s work has been presented here as a hyper-refinement of the modernist imagery. The computation is entirely visual. A reassembly of the layered symmetries explains the structure of the symmetry of the house and provides an illustration of the basic thesis of this research on the foundation of a theory of emergence based on symmetry considerations. All plans of the house are represented in three different levels of abstraction moving successively away from the architectural representation to a purely diagrammatic one that foregrounds divisions of space. All representations are fed into an analysis algorithm to pick up all symmetry relationships and the parts are constructed as instances of a binary composition of a family of rectangular grids. Finally the process is reversed to fully account for the construction of the space of the house as a three dimensional layered composition. Lastly, this research points to two categories of extension; a) on the improvement of the system itself; and b) on the interpretative capabilities it affords for the construction and evaluation of critical languages of design.

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Thesis (Ph. D.)--State University of New York at Binghamton, of Department of Mathematical Sciences, 2007. / Includes bibliographical references.

Thesis (M. Sc.)--Carleton University, 2002. / Includes bibliographical references (p. 37-38). Also available in electronic format on the Internet.

Thesis (M.S.)--Ohio University, August, 1997. / Title from PDF t.p.