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In this thesis we consider in detail the following two problems for semigroups: (i) When are semigroups finitely generated and presented? (ii) Which families of semigroups can be efficiently presented? We also consider some other finiteness conditions for semigroups, homology of semigroups and wreath product of groups. In Chapter 2 we investigate finite presentability and some other finiteness conditions for the O-direct union of semigroups with zero. In Chapter 3 we investigate finite generation and presentability of Rees matrix semigroups over semigroups. We find necessary and sufficient conditions for finite generation and presentability. In Chapter 4 we investigate some other finiteness conditions for Rees matrix semigroups. In Chapter 5 we consider groups as semigroups and investigate their semigroup efficiency. In Chapter 6 we look at "proper" semigroups, that is semigroups that are not groups. We first give examples of efficient and inefficient "proper" semigroups by computing their homology and finding their minimal presentations. In Chapter 7 we compute the second homology of finite simple semigroups and find a "small" presentation for them. If that "small" presentation has a special relation, we prove that finite simple semigroups are efficient. Finally, in Chapter 8, we investigate the efficiency of wreath products of finite groups as groups and as semigroups. We give more examples of efficient groups and inefficient groups.

<b>Chapter One</b> After the definitions and basic results required for the rest of the thesis, a notion of spectrum for semigroup representations is introduced and some relevant examples given. <b>Chapter Two</b> Any semigroup representation by isometries on a Banach space may be dilated to a group representation on a larger Banach space. A new proof of this result is presented here, and a connection is shown to exist between the dilation and the trajectories of the dual representation. The problem of dilating various types of spaces, including partially ordered spaces, C*-algebras, and reflexive spaces, is discussed, and new dilation theorems are given for dual Banach spaces and von Neumann algebras. <b>Chapter Three</b> In this chapter the spectrum of a representation is examined more closely with the aid of methods from Banach algebra theory. In the case where the representation is by isometries it is shown that the spectrum is non-empty, that it is compact if and only if the representation is norm-continuous, and that any isolated point in the unitary spectrum is an eigenvalue. <b>Chapter Four</b> An analytic characterisation is given of the spectral conditions that imply a representation by isometries is invertible. For representations of Z+_n this con- dition is shown to be equivalent to polynomial convexity. Some topological conditions on the spectrum are also shown to imply invertibility. <b>Chapter Five</b> The ideas of the previous chapters are applied to problems of asymptotic behaviour. Asymptotic stability is described in terms of the behaviour of the dual of a representation. Finally, the case when the unitary spectrum is countable is discussed in detail.

This thesis is a study of semitopological groups, a similar but weaker notion than that of topological groups. It is shown that all topological groups are semitopological groups but that the converse is not true. This thesis investigates some of the conditions under which semitopological groups are, in fact, topological groups. It is assumed that the reader is familiar with basic group theory and topology.

This thesis describes properties of Abelian groups, and develops a study of the properties of divisibility in Abelian groups.

An abundance of information regarding the structure of a finite group can be obtained by studying its irreducible characters. Of particular interest are monomial characters – those induced from a linear character of some subgroup – since Brauer has shown that any irreducible character of a group can be written as an integral linear combination of monomial characters. Our primary focus is the class of M-groups, those groups all of whose irreducible characters are monomial. A classical theorem of Taketa asserts that an M-group is necessarily solvable, and Dade proved that every solvable group can be embedded as a subgroup of an M-group. After discussing results related to M-groups, we will construct explicit families of solvable groups that cannot be embedded as subnormal subgroups of any M-group. We also discuss groups possessing a unique non-monomial irreducible character, and prove that such a group cannot be simple.

The main objective of this thesis was to find the full automorphism groups of finite Desarguesian planes. A set of homologies were used to generate the automorphism group when the order of the plane was prime. When the order was a prime power Pa,a ≠ 1 the Frobenius automorphism was added to the set of homologies, and then the full automorphism group was generated. The Frobenius automorphism was found by using the planar ternary ring derived from a coordinatization of the plane. / Includes bibliography. / Thesis (M.S.)--Florida Atlantic University, 2013.

by Tang Ko Cheung, Simon. / Thesis date on t.p. originally printed as 1997, of which 7 has been overwritten as 8 to become 1998. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1998. / Includes bibliographical references (leaves 95-97). / Chapter 1 --- Introduction --- p.5 / Chapter 1.1 --- Motivation --- p.5 / Chapter 1.1.1 --- Calculation of the Galois group --- p.5 / Chapter 1.1.2 --- Factorization of polynomials in a finite number of steps IS feasible --- p.6 / Chapter 1.2 --- Table & Diagram of Transitive Groups up to Degree 7 --- p.8 / Chapter 1.3 --- Background and Notation --- p.13 / Chapter 1.4 --- Content and Contribution of THIS thesis --- p.17 / Chapter 2 --- Stauduhar's Method --- p.20 / Chapter 2.1 --- Overview & Restrictions --- p.20 / Chapter 2.2 --- Representation of the Galois Group --- p.21 / Chapter 2.3 --- Groups and Functions --- p.22 / Chapter 2.4 --- Relative Resolvents --- p.24 / Chapter 2.4.1 --- Computing Resolvents Numerically --- p.24 / Chapter 2.4.2 --- Integer Roots of Resolvent Polynomials --- p.25 / Chapter 2.5 --- The Determination of Galois Groups --- p.26 / Chapter 2.5.1 --- Searching Procedures --- p.26 / Chapter 2.5.2 --- "Data: T(x1,x2 ,... ,xn), Coset Rcpresentatives & Searching Diagram" --- p.27 / Chapter 2.5.3 --- Examples --- p.32 / Chapter 2.6 --- Quadratic Factors of Resolvents --- p.35 / Chapter 2.7 --- Comment --- p.35 / Chapter 3 --- Factoring Polynomials Quickly --- p.37 / Chapter 3.1 --- History --- p.37 / Chapter 3.1.1 --- From Feasibility to Fast Algorithms --- p.37 / Chapter 3.1.2 --- Implementations on Computer Algebra Systems --- p.42 / Chapter 3.2 --- Squarefree factorization --- p.44 / Chapter 3.3 --- Factorization over finite fields --- p.47 / Chapter 3.4 --- Factorization over the integers --- p.50 / Chapter 3.5 --- Factorization over algebraic extension fields --- p.55 / Chapter 3.5.1 --- Reduction of the problem to the ground field --- p.55 / Chapter 3.5.2 --- Computation of primitive elements for multiple field extensions --- p.58 / Chapter 4 --- Soicher-McKay's Method --- p.60 / Chapter 4.1 --- "Overview, Restrictions and Background" --- p.60 / Chapter 4.2 --- Determining cycle types in GalQ(f) --- p.62 / Chapter 4.3 --- Absolute Resolvents --- p.64 / Chapter 4.3.1 --- Construction of resolvent --- p.64 / Chapter 4.3.2 --- Complete Factorization of Resolvent --- p.65 / Chapter 4.4 --- Linear Resolvent Polynomials --- p.67 / Chapter 4.4.1 --- r-sets and r-sequences --- p.67 / Chapter 4.4.2 --- Data: Orbit-length Partitions --- p.68 / Chapter 4.4.3 --- Constructing Linear Resolvents Symbolically --- p.70 / Chapter 4.4.4 --- Examples --- p.72 / Chapter 4.5 --- Further techniques --- p.72 / Chapter 4.5.1 --- Quadratic Resolvents --- p.73 / Chapter 4.5.2 --- Factorization over Q(diac(f)) --- p.73 / Chapter 4.6 --- Application to the Inverse Galois Problem --- p.74 / Chapter 4.7 --- Comment --- p.77 / Chapter A --- Demonstration of the MAPLE program --- p.78 / Chapter B --- Avenues for Further Exploration --- p.84 / Chapter B.1 --- Computational Galois Theory --- p.84 / Chapter B.2 --- Notes on SAC´ؤSymbolic and Algebraic Computation --- p.88 / Bibliography --- p.97

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The compact symmetric multicategories (CSMs) introduced by Joyal and Kock in their 2011 note 'Feynman Graphs, and Nerve Theorem for Compact Symmetric Multicategories' [JK11] directly generalise a number of unital operad types, such as wheeled properads, that admit a contraction operation as well as an operadic multiplication. These structures are known to exhibit strange behaviour related to the contraction of units, and this is problematic for [JK11]. In this thesis, I modify the construction of [JK11] to obtain non unital (coloured) modular operads as algebras for a monad defined in terms of connected graphs, and use this as a foundation for a new construction of CSMs based on special graph morphisms. A corresponding nerve theorem characterises CSMs in terms of a Segal condition. This construction sheds light, and provides some control, on the behaviour of the contracted units.