Thesis (Ph. D.)--University of Texas at Austin, 1998. / Vita. Includes bibliographical references (leaves 61-63). Available also in a digital version from Dissertation Abstracts.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1999. / Includes bibliographical references. Also available on the Internet.

Algebraic theories describe mathematical structures that are defined in terms of operations and equations, and are extremely important throughout mathematics. Many generalisations of the classical notion of an algebraic theory have sprung up for use in different mathematical contexts; some examples include Lawvere theories, monads, PROPs and operads. The first central notion of this thesis is a common generalisation of these, which we call a proto-theory. The purpose of an algebraic theory is to describe its models, which are structures in which each of the abstract operations of the theory is given a concrete interpretation such that the equations of the theory hold. The process of going from a theory to its models is called semantics, and is encapsulated in a semantics functor. In order to define a model of a theory in a given category, it is necessary to have some structure that relates the arities of the operations in the theory with the objects of the category. This leads to the second central notion of this thesis, that of an interpretation of arities, or aritation for short. We show that any aritation gives rise to a semantics functor from the appropriate category of proto-theories, and that this functor has a left adjoint called the structure functor, giving rise to a structure{semantics adjunction. Furthermore, we show that the usual semantics for many existing notions of algebraic theory arises in this way by choosing an appropriate aritation. Another aim of this thesis is to find a convenient category of monads in the following sense. Every right adjoint into a category gives rise to a monad on that category, and in fact some functors that are not right adjoints do too, namely their codensity monads. This is the structure part of the structure{semantics adjunction for monads. However, the fact that not every functor has a codensity monad means that the structure functor is not defined on the category of all functors into the base category, but only on a full subcategory of it. This deficiency is solved when passing to general proto-theories with a canonical choice of aritation whose structure{semantics adjunction restricts to the usual one for monads. However, this comes at a cost: the semantics functor for general proto-theories is not full and faithful, unlike the one for monads. The condition that a semantics functor be full and faithful can be thought of as a kind of completeness theorem | it says that no information is lost when passing from a theory to its models. It is therefore desirable to retain this property of the semantics of monads if possible. The goal then, is to find a notion of algebraic theory that generalises monads for which the semantics functor is full and faithful with a left adjoint; equivalently the semantics functor should exhibit the category of theories as a re ective subcategory of the category of all functors into the base category. We achieve this (for well-behaved base categories) with a special kind of proto-theory enriched in topological spaces, which we call a complete topological proto-theory. We also pursue an analogy between the theory of proto-theories and that of groups. Under this analogy, monads correspond to finite groups, and complete topological proto-theories correspond to profinite groups. We give several characterisations of complete topological proto-theories in terms of monads, mirroring characterisations of profinite groups in terms of finite groups.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / O propÃsito desta dissertaÃÃo à apresentar aplicaÃÃes da Teoria dos Grafos à Teoria dos Grupos. De posse do grafo associado a um grupo finito, nÃs obtemos vÃrios resultados interessantes sobre a estrutura do grupo analisando tal grafo à luz de tÃcnicas-padrÃo da Teoria dos Grafos. Mais precisamente, os nÃmeros cromÃtico e de independÃncia do grafo de um grupo finito nos permitem estimar a cardinalidade mÃxima de um subgrupo abeliano do mesmo, bem como o tamanho mÃnimo possÃvel de um subconjunto do grupo formado por elementos que nÃo comutam dois a dois; no caso de grupos finitos abelianos, nÃs tambÃm estudamos seus subconjuntos livres de somas. / This report deals with applications of Graph Theory to Group Theory. Once we construct the graph associated to a finite group, we get several interesting results on the group structure by analysing its associated graph with the help of various standard graph-theoretic tools. More precisely, the chromatic and independence numbers of the graph of a finite group allows us to estimate the maximal cardinality of an abelian subgroup of it, as well as the minimal size of a subset of the group, all of whose elements donât commute in pairs; for finite abelian groups, we also study their free-sum subsets.

The idea of keychains, an (n+1)-tuple of non-increasing real numbers in the unit interval always including 1, naturally arises in study of finite fuzzy set theory. They are a useful concept in modeling ideas of uncertainty especially those that arise in Economics, Social Sciences, Statistics and other subjects. In this thesis we define and study some basic properties of keychains with reference to Partially Ordered Sets, Lattices, Chains and Finite Fuzzy Sets. We then examine the role of keychains and their lattice diagrams in representing uncertainties that arise in such problems as in preferential voting patterns, outcomes of competitions and in Economics - Preference Relations.

We determine the number and nature of distinct equivalence classes of fuzzy subgroups of finite Abelian p-group G of rank two under a natural equivalence relation on fuzzy subgroups. Our discussions embrace the necessary theory from groups with special emphasis on finite p-groups as a step towards the classification of crisp subgroups as well as maximal chains of subgroups. Unique naming of subgroup generators as discussed in this work facilitates counting of subgroups and chains of subgroups from subgroup lattices of the groups. We cover aspects of fuzzy theory including fuzzy (homo-) isomorphism together with operations on fuzzy subgroups. The equivalence characterization as discussed here is finer than isomorphism. We introduce the theory of keychains with a view towards the enumeration of maximal chains as well as fuzzy subgroups under the equivalence relation mentioned above. We discuss a strategy to develop subgroup lattices of the groups used in the discussion, and give examples for specific cases of prime p and positive integers n,m. We derive formulas for both the number of maximal chains as well as the number of distinct equivalence classes of fuzzy subgroups. The results are in the form of polynomials in p (known in the literature as Hall polynomials) with combinatorial coefficients. Finally we give a brief investigation of the results from a graph-theoretic point of view. We view the subgroup lattices of these groups as simple, connected, symmetric graphs.

In this thesis we use the natural equivalence of fuzzy subgroups studied by Murali and Makamba [25] to characterize fuzzy subgroups of some finite groups. We focus on the determination of the number of equivalence classes of fuzzy subgroups of some selected finite groups using this equivalence relation and its extension. Firstly we give a brief discussion on the theory of fuzzy sets and fuzzy subgroups. We prove a few properties of fuzzy sets and fuzzy subgroups. We then introduce the selected groups namely the symmetric group 3 S , dihedral group 4 D , the quaternion group Q8 , cyclic p-group pn G = Z/ , pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We also present their subgroups structures and construct lattice diagrams of subgroups in order to study their maximal chains. We compute the number of maximal chains and give a brief explanation on how the maximal chains are used in the determination of the number of equivalence classes of fuzzy subgroups. In determining the number of equivalence classes of fuzzy subgroups of a group, we first list down all the maximal chains of the group. Secondly we pick any maximal chain and compute the number of distinct fuzzy subgroups represented by that maximal chain, expressing each fuzzy subgroup in the form of a keychain. Thereafter we pick the next maximal chain and count the number of equivalence classes of fuzzy subgroups not counted in the first chain. We proceed inductively until all the maximal chains have been exhausted. The total number of fuzzy subgroups obtained in all the maximal chains represents the number of equivalence classes of fuzzy subgroups for the entire group, (see sections 3.2.1, 3.2.2, 3.2.6, 3.2.8, 3.2.9, 3.2.15, 3.16 and 3.17 for the case of selected finite groups). We study, establish and prove the formulae for the number of maximal chains for the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . To accomplish this, we use lattice diagrams of subgroups of these groups to identify the maximal chains. For instance, the group pn qm G = Z/ + Z/ would require the use of a 2- dimensional rectangular diagram (see section 3.2.18 and 5.3.5), while for the group pn qm r s G = Z/ + Z/ + Z/ we execute 3- dimensional lattice diagrams of subgroups (see section 5.4.2, 5.4.3, 5.4.4, 5.4.5 and 5.4.6). It is through these lattice diagrams that we identify routes through which to carry out the extensions. Since fuzzy subgroups represented by maximal chains are viewed as keychains, we give a brief discussion on the notion of keychains, pins and their extensions. We present propositions and proofs on why this counting technique is justifiable. We derive and prove formulae for the number of equivalence classes of the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We give a detailed explanation and illustrations on how this keychain extension principle works in Chapter Five. We conclude by giving specific illustrations on how we compute the number of equivalence classes of a fuzzy subgroup for the group p2 q2 r 2 G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of the group p q r G = Z/ + Z/ + Z/ 1 2 2 . This illustrates a general technique of computing the number of fuzzy subgroups of G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of 1 -1 = / + / + / pn qm r s G Z Z Z . Our illustration also shows two ways of extending from a lattice diagram of 1 G to that of G .

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Tesis por compendio / [ES] Un problema clásico en la teoría de grupos finitos es el estudio de cómo los tamaños de las clases de conjugación influyen sobre la estructura del grupo. En las últimas décadas, numerosos investigadores han obtenido nuevos avances en esta línea. Especialmente, se han probado resultados interesantes a partir de la información proporcionada por los tamaños de clase de algún subconjunto de elementos del grupo, como los elementos de orden potencia de primo, elementos p-regulares, etc. Además, ciertos subconjuntos de elementos definidos a través de la tabla de caracteres del grupo están siendo investigados recientemente, como los elementos anuladores y los elementos reales. Por otra parte, en los últimos años, el estudio de grupos factorizados como producto de subgrupos ha sido objeto de creciente interés. En particular, diversos autores han analizado la estructura de grupos factorizados en los que diferentes familias de subgrupos de los factores satisfacen ciertas condiciones de permutabilidad. En esta tesis pretendemos conjugar ambas perspectivas de actualidad en la teoría de grupos de manera novedosa. Así, en este contexto de literatura escasa, el objetivo es obtener nuevas contribuciones acerca de la estructura global de un grupo factorizado a partir de ciertas propiedades aritméticas de los tamaños de las clases de algunos elementos de sus factores. Estudiamos productos de dos subgrupos, eventualmente mutuamente permutables, donde los elementos (p-regulares) de orden potencia de primo de los factores tienen tamaños de clase libres de cuadrados. Analizamos el caso de tamaños de clase potencias de primos para grupos factorizados arbitrarios, evitando el uso de condiciones de permutabilidad entre los factores involucrados. El concepto de una core-factorización de un grupo, que extiende en particular a los productos mutuamente permutables, es introducido por primera vez en esta tesis y ha resultado crucial dentro de este contexto. Esta noción surge precisamente cuando consideramos las anteriores propiedades aritméticas para los tamaños de clase de elementos anuladores, interrelacionando novedosamente la teoría de caracteres con la investigación en grupos factorizados. Finalmente, estudiamos grupos que poseen una core-factorización cuyos tamaños de clase de pi-elementos (de orden potencia de primo) son pi-números o pi'-números. / [CA] Un problema clàssic dins de la teoria de grups finits és l'estudi de com els tamanys de les classes de conjugació influeixen sobre l'estructura del grup. En les últimes dècades, nombrosos investigadors han obtingut nous avanços en aquesta línia. Especialment, s'han provat resultats interessants a partir de la informació proporcionada pels tamanys de classe d'algun subconjunt d'elements del grup, com els elements d'ordre potència de primer, elements p-regulars, etc. A més, certs subconjunts d'elements definits a través de la taula de caràcters del grup estan sent investigats recentment, com els elements anul·ladors i els elements reals. D'altra banda, en els últims anys, l'estudi de grups factoritzats com a producte de subgrups ha sigut objecte de creixent interés. En particular, diversos autors han analitzat l'estructura de grups factoritzats en els quals diferents famílies de subgrups dels factors satisfan certes condicions de permutabilitat. En aquesta tesi pretenem conjugar ambdues perspectives d'actualitat en la teoria de grups de manera innovadora. En aquest context de literatura escassa, l'objectiu és obtenir noves contribucions sobre l'estructura global d'un grup factoritzat a partir de certes propietats aritmètiques dels tamanys de les classes d'alguns elements dels seus factors. Estudiem productes de dos subgrups, eventualment mútuament permutables, on els elements (p-regulars) d'ordre potència de primer dels factors tenen tamany de classe llibre de quadrats. Analitzem el cas de tamanys de classe potències de primers per a grups factoritzats arbitraris, evitant l'ús de condicions de permutabilitat entre els factors involucrats. El concepte d'una core-factorització d'un grup, que estén particularment als productes mútuament permutables, és introduït per primera vegada en aquesta tesi i ha resultat determinant dins d'aquest context. Aquesta noció sorgeix precisament quan considerem les propietats aritmètiques anteriors per als tamanys de classe d'elements anul·ladors, interrelacionant innovadorament la teoria de caràcters amb la investigació en grups factoritzats. Finalment, estudiem grups els quals posseeixen una core-factorització on els tamanys de classe dels pi-elements (d'ordre potència de primer) són pi-números o pi'-números. / [EN] The influence of the conjugacy class sizes on the structure of a group has been a widely investigated problem within finite group theory. In the last decades, several researchers have obtained new progress in this direction. Specially, some relevant information is provided by the class sizes of certain subsets of elements of the group, as prime power order elements, p-regular elements, etc. Other subsets of elements that have recently attracted interest are defined via the character table of the group, as vanishing elements and real elements. In parallel to this research on conjugacy classes, the study of groups which can be factorised as a product of two subgroups has gained increasing interest. In particular, the structure of factorised groups such that different families of subgroups of the factors satisfy certain permutability conditions has recently been analysed. In this thesis we aim to combine in a novel way both perspectives of group theory. In this framework of very scarce literature, our main purpose is to obtain new contributions about the global structure of a factorised group when the class lengths of some elements in its factors verify certain arithmetical properties. Square-free class length conditions on (p-regular) prime power order elements are considered for products of two subgroups, occasionally mutually permutable. Prime power class sizes are investigated for arbitrary products of two groups, avoiding the use of permutability conditions between the factors. The concept of a core-factorisation of a group, which particularly extends products of mutually permutable subgroups, is introduced for the first time in this dissertation, and it has been revealed determinant within this context. Precisely, this notion emerges when discussing the above arithmetical properties on the class sizes of vanishing elements, interplaying as a novelty character theory and the research on factorised groups. Core-factorisations are also exploited when analysing pi-number and pi'-number class lengths for (prime power order) pi-elements in the factors of a factorised group. / This dissertation has been elaborated at the Instituto Universitario de Matemática Pura y Aplicada de la Universitat Politècnica de València (IUMPA-UPV), thanks mainly to the financial support of the predoctoral grant ACIF/2016/170 from Generalitat Valenciana (Spain). The first academic year was supported by Proyecto Prometeo II/2013/013 from Generalitat Valenciana. The institute IUMPA has financed some travel expenses of the author’s attendances to research conferences. This research has been partially supported by Proyecto PGC2018-096872-B-I00, Ministerio de Ciencia, Innovación y Universidades. The mobility grant BEFPI/2018/025 from Generalitat Valenciana has allowed the author to perform a research stay of three months (March-May 2018) at the Dipartimento di Matematica e Informatica “U. Dini” (DIMAI) of Università di Firenze (Italy). The author has also been granted with a Borsa Ferran Sunyer i Balaguer for a research stay at Università di Firenze in April 2019. / Ortiz Sotomayor, VM. (2019). Conjugacy classes and factorised groups [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/125710 / Compendio