Thesis (Ph. D.)--University of Texas at Austin, 2004. / Supervisors: Willy Fischler and Yuval Neʼeman. Vita. Includes bibliographical references.

Thesis (Ph. D.)--Ohio State University, 2005. / Title from first page of PDF file. Document formatted into pages; contains viii, 87 p. Includes bibliographical references (p. 85-87). Available online via OhioLINK's ETD Center

A group whose co-word problem is a context free language is called coCF. Lehnert's conjecture states that a group G is coCF if and only if G embeds as a finitely generated subgroup of R. Thompson's group V. In this thesis we explore a class of groups, Faug, proposed by Berns-Zieze, Fry, Gillings, Hoganson, and Mathews to contain potential counterexamples to Lehnert's conjecture. We create infinite and finite presentations for such groups and go on to prove that a certain subclass of Faug consists of groups that do embed into V. By Anisimov a group has regular word problem if and only if it is finite. It is also known that a group G is finite if and only if there exists an embedding of G into V such that its natural action on C₂:= {0,1}^w is free on the whole space. We show that the class of groups with a context free word problem, the class of CF groups, is precisely the class of finitely generated demonstrable groups for V. A demonstrable group for V is a group G which is isomorphic to a subgroup in V whose natural action on C₂ acts freely on an open subset. Thus our result extends the correspondence between language theoretic properties of groups and dynamical properties of subgroups of V. Additionally, our result also shows that the final condition of the four known closure properties of the class of coCF groups also holds for the set of finitely generated subgroups of V.

We study finite generation, 2-generation and simplicity of subgroups of H[sub]c, the group of homeomorphisms of Cantor space. In Chapter 1 and Chapter 2 we run through foundational concepts and notation. In Chapter 3 we study vigorous subgroups of H[sub]c. A subgroup G of H[sub]c is vigorous if for any non-empty clopen set A with proper non-empty clopen subsets B and C there exists g ∈ G with supp(g) ⊑ A and Bg ⊆ C. It is a corollary of the main theorem of Chapter 3 that all finitely generated simple vigorous subgroups of H[sub]c are in fact 2-generated. We show the family of finitely generated, simple, vigorous subgroups of H[sub]c is closed under several natural constructions. In Chapter 4 we use a generalised notion of word and the tight completion construction from [13] to construct a family of subgroups of H[sub]c which generalise Thompson's group V . We give necessary conditions for these groups to be finitely generated and simple. Of these we show which are vigorous. Finally we give some examples.

We begin our study of amalgamations by examining some ideas which are well-known for the category of R-modules. In particular we look at such notions as direct limits, pushouts, pullbacks, tensor products and flatness in the category of S-sets. Chapter II introduces the important concept of free extensions and uses this to describe the amalgamated free product. In Chapter III we define the extension property and the notion of purity. We show that many of the important notions in semigroup amalgams are intimately connected to these. In Section 2 we deduce that 'the extension property implies amalgamation' and more surprisingly that a semigroup U is an amalgamation base if and only if it has the extension property in every containing semigroup. Chapter IV revisits the idea of flatness and after some technical results we prove a result, similar to one for rings, on flat amalgams. In Chapter V we show that the results of Hall and Howie on perfect amalgams can be proved using the same techniques as those used in Chapters III and IV. We conclude the thesis with a look at the case of rings. We show that almost all of the results for semi group amalgams examined in the previous chapters, also hold for ring amalgams.

The concept of boundaries in group theory gained prominence in the 70s and 80s mainly as a construct to describe significant group events. A contributing factor was when general systems theory, in which boundaries are central, was applied to living systems. Boundaries continued to be used predominantly to refer to structural aspects of a group, such as time structuring, membership, role, subgroupings, and task, and, to a lesser extent, as an abstract construct to refer to group processes and dynamics. In group practice, the use of boundaries as a guide and instrument to gauge group dynamics has been limited. In general, boundaries are not used to assess group events in order to determine a course of action or intervention. The first part of the research explores the concept of boundaries in three theoretical frameworks. The second part of the research explores the application of boundaries as a construct central to the understanding of group dynamics in an experiential time-limited training group. It also examines ways in which this can lead to enhanced group practice. The focus was on boundaries as psychological dimensions in the group space. In the exploration of boundaries in existing theoretical frameworks, an important link between boundaries and trauma, which inevitably involves a breach and violation of boundaries, was highlighted. A novel qualitative content analysis method was designed to reveal boundary changes systematically and to show how boundaries were redefined over a period of time. A unique feature of this computer assisted (Atlas.ti) method is that boundary shifts are quantitatively tracked, allowing further qualitative exploration. This method was applied in a case study of a training group, so demonstrating the applicability of the method to the study of small groups. Results of the case study revealed the impact that events prior the group had on group boundary development, in particular emotional linking in the group. Shifts in psychological boundaries were clearly visible in the quantitative analysis of boundaries in focus, across boundaries, indicated by transactions across boundaries. South Africa, as is the case in other societies in transition, is characterised by continuous breaches and violations of boundaries. By viewing group interactions through a boundary lens, group leaders can understand the complexity of group dynamics better. With this understanding, facilitators and leaders of groups can deliberately influence psychological boundaries. In so doing they can create opportunities for individual transitions and societal transformation. / Thesis (PhD)--University of Pretoria, 2013. / gm2014 / Psychology / Unrestricted

<p> We present a discussion of the application of group theory to the particular case of solid methane, in all its crystalline phases. </p> <p> We also employ the quantum mechanical mean approximation to derive the mean square angle of deviation of the methane free molecule. By means of group theory we derive the normal modes, the symmetry coordinates and the nuclear spin functions of methane, which may be found useful for many other purposes in the study of methane. Finally, using these results, we give a discussion of the infrared and Raman spectra based on group theory again, to explain the observed transitions of the methane molecule in its condensed phases. We conclude that the λ type transitional are caused by changes in molecular orientation. Phase I is probably disordered,while phase II has structure of symmetry D2d. Phase III (of CD4) is ordered but of lower symmetry and unclear structure. </p> <p> A possible explanation probably requires an arrangement having more molecules per unit cell than in phases I and II. </p> / Thesis / Master of Science (MSc)

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