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In this paper the following conjecture is studied: the group von Neumann algebra N(G) is a flat CG-module if and only if the group G is locally virtually cyclic. This paper proves that if G is locally virtually cyclic, then N(G) is flat as a CG-module. The converse is proved for the class of all elementary amenable groups without infinite locally finite subgroups. Foundational cases for which the conjecture is shown to be true are the groups G=Z, G=ZxZ, G=Z*Z, Baumslag-Solitar groups, and some infinitely-presented variations of Baumslag-Solitar groups. Modules other than N(G), such as L^p-spaces and group C*-algebras, are considered as well. The primary tool that is used to achieve many of these results is group homology. / Ph. D.

This paper is concerned with a generalization of the concept of unitary equivalence of spectral measures on a Baer *-semigroup. A connection is made between abstract spectral measures, and three other distinct mathematical systems. Chapter II is devoted specifically to generalizing the concept of a spectral measure and to determining necessary and sufficient conditions for which two spectral measures will be unitarily equivalent. Chapter III discusses the problem of each (C(M) , qμ) being type I in terms of cycles, the basic elements of C(M). In Chapter IV it is shown that in a Loomis *-semigroup each type I (C(M) , qμ) will be type I homogeneous. Chapter V relates the study of unitary equivalence of spectral measures and the unitary equivalence of normal elements in a Finite Dimensional Baer *-algebra. / Ph. D.

Topological groupoids with"approximate" inverses are studied. In the compact case, these"approximate" inverses turn out to be true inverses. Examples of groupoids wL:h"approximate" inverses are given in the section dealing with function spaces. Using the classical construction of Haar as a guide, we succeed in obtaining a (non-trivial) regular, right-invariant measure over a locally compact left group satisfying the conditions: a) open sets are preserved by left translation b) each group component is open. In the section dealing with integrals, we consider a compact metric topological semigroup that is right simple 3nd possesses a right contractive metric (ρ(xz,yz) ≤ ρ(x,y)). It is shown that such a structure always carries a non-trivial right-invariant integral. Throughout the entire development, associativity is invoked only once. The investigation concludes with a section dealing with sufficient conditions under which binary-topological systems become topological groups. A mob-group is defined to be a T_o-space which is also an algebraic group. A theorem in the last section states that a mob-group is a topological group iff given any open set W about the identity, W ∩ W⁻¹ has non-void interior. / Ph. D.

This thesis examines the algebraic structure of the Rubik’s Cube—focusing on both the classic 3×3×3 model and its generalization to an n×n×n model—through the application of group theory. It delineates the fundamental group-theoretic characterizations of the Rubik’s Cube and establishes necessary and sufficient conditions for its solvability. Utilizing these conditions, formulas are derived for the number of solvable configurations of the Rubik’s Cube across all sizes.

In this thesis, we mainly address two contrasting topics in the area of saturated fusion systems. The first concerns the notion of a centraliser of a subsystem E of a fusion system F, and we give new proofs of the existence of such an object in the case where E is normal in F. The second concerns the development of the theory of `trees of fusion systems', an analogue for fusion systems of Bass-Serre theory for finite groups. A major theorem finds conditions on a tree of fusion systems for there to exist a saturated completion, and this is applied to construct and classify certain fusion systems over p-groups with an abelian subgroup of index p. Results which do not fall into either of the above categories include a new proof of Thompson's normal p-complement Theorem for saturated fusion systems and characterisations of certain quotients of fusion systems which possess a normal subgroup.