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A long-standing conjecture states that if A is a finite group acting fixed point freely on a finite solvable group G of order coprime to jAj, then the Fitting length of G is bounded by the length of the longest chain of subgroups of A. If A is nilpotent, it is expected that the conjecture is true without the coprimeness condition. We prove that the conjecture without the coprimeness condition is true when A is a cyclic group whose order is a product of three primes which are coprime to 6 and the Sylow 2-subgroups of G are abelian. We also prove that the conjecture without the coprimeness condition is true when A is an abelian group whose order is a product of three primes which are coprime to 6 and jGj is odd.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: As an introduction to point-free topology, we will explicitly show the connection between topology and frames (locales) and introduce an abstract notion, which in the point-free setting, can be thought of as a subspace of a topological space. In this setting, we refer to this notion as a sublocale and we will show that there are at least four ways to represent sublocales. By using the language of category theory, we proceed by investigating closure in the point-free setting by way of operators. We de ne what we mean by a coclosure operator in an abstract context and give two seemingly di erent examples of co-closure operators of Frm. These two examples are then proven to be the same. Compactness is one of the most important notions in classical topology and therefore one will nd a great number of results obtained on the subject. We will undertake a study into the interrelationship between three weaker compact notions, i.e. feeble compactness, pseudocompactness and countable compactness. This relationship has been established and is well understood in topology, but (to a degree) the same cannot be said for the point-free setting. We will give the frame interpretation of these weaker compact notions and establish a point-free connection. A potentially promising result will also be mentioned. / AFRIKAANSE OPSOMMING: As 'n inleiding tot punt-vrye topologie, sal ons eksplisiet die uiteensetting van hierdie benadering tot topologie weergee. Ons de nieer 'n abstrakte konsep wat, in die punt-vrye konteks, ooreenstem met 'n subruimte van 'n topologiese ruimte. Daar sal verder vier voorstellings van hierdie konsep gegee word. Afsluiting, deur middel van operatore, word in die puntvrye konteks ondersoek met behulp van kategorie teorie as taalmedium. Ons sal 'n spesi eke operator in 'n abstrakte konteks de nieer en twee o enskynlik verskillende voorbeelde van hierdie operator verskaf. Daar word dan bewys dat hierdie twee operatore dieselfde is. Kompaktheid is een van die mees belangrikste konsepte in klassieke topologie en as gevolg daarvan geniet dit groot belangstelling onder wiskundiges. 'n Studie in die verwantskap tussen drie swakker forme van kompaktheid word onderneem. Hierdie verwantskap is al in topologie bevestig en goed begryp onder wiskundiges. Dieselfde kan egter, tot 'n mate, nie van die puntvrye konteks ges^e word nie. Ons sal die puntvrye formulering van hierdie swakker konsepte van kompaktheid en hul verbintenis, weergee. 'n Resultaat wat moontlik belowend kan wees, sal ook genoem word.

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In this thesis we present the Stone representation theorem, generally known as Stone duality in the point-free context. The proof is choice-free and, since we do not have to be concerned with points, it is by far simpler than the original. For each infinite cardinal κ we show that the counter- part of the κ-complete Boolean algebras is constituted by the κ-basically disconnected Stone frames. We also present a precise characterization of the morphisms which correspond to the κ-complete Boolean homomorphisms. Although Booleanization is not functorial in general, in the part of the dual- ity for extremally disconnected Stone frames it is, and constitutes an equiv- alence of categories. We finish the thesis by focusing on the De Morgan (or extremally disconnected) frames and present a new characterization of these by their superdense sublocales. We also show that in contrast with this phenomenon, a metrizable frame has no non-trivial superdense sublocale; in other words, a non-trivial Čech-Stone compactification of a metrizable frame is never metrizable. 1

1 This thesis concerns separation axioms in point-free topology. We introduce a notion of weak inclusion, which is a relation on a frame that is weaker than the relation ≤. Weak inclusions provide a uniform way to work with standard separation axioms such as subfitness, fitness, and regularity. Proofs using weak inclusions often bring new insight into the nature of the axioms. We focus on results related to the axiom of subfitness. We study a sublocale which is defined as the intersection of all the codense sublocales of a frame. We show that it need not be subfit. For spacial frames, it need not be spacial.

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Usability aspects of programming languages are often overlooked, yet have a substantial effect on programmer productivity. These issues are even more acute in the field of Inductive Synthesis, where programs are automatically generated from sample expected input and output data, and the programmer needs to be able to comprehend, and confirm or reject the suggested programs. A promising method of Inductive Synthesis, CombInduce, which is particularly suitable for synthesizing recursive programs, is a candidate for improvements in usability as the target language Combilog is not user-friendly. The method requires the target language to be strictly compositional, hence devoid of variables, yet have the expressiveness of definite clause programs. This sets up a challenging problem for establishing a user-friendly but equally expressive target language. Alternatives to Combilog, such as Quine's Predicate-functor Logic and Schönfinkel and Curry's Combinatory Logic also do not offer a practical notation: finding a more usable representation is imperative. This thesis presents two distinct approaches towards more convenient representations which still maintain compositionality. The first is Visual Combilog (VC), a system for visualizing Combilog programs. In this approach Combilog remains as the target language for synthesis, but programs can be read and modified by interacting with the equivalent diagrams instead. VC is implemented as a split-view editor that maintains the equivalent Combilog and VC representations on-the-fly, automatically transforming them as necessary. The second approach is Combilog with Name Projection (CNP), a textual iteration of Combilog that replaces numeric argument positions with argument names. The result is a language where argument names make the notation more readable, yet compositionality is preserved by avoiding variables. Compositionality is demonstrated by implementing CombInduce with CNP as the target language, revealing that programs with the same level of recursive complexity can be synthesized in CNP equally well, and establishing the underlying method of synthesis can also work with CNP. Our evaluations of the user-friendliness of both representations are supported by a range of methods from Information Visualization, Cognitive Modelling, and Human-Computer Interaction. The increased usability of both representations are confirmed by empirical user studies: an often neglected aspect of language design.