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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]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/125710 / Compendio

It can be deduced from the Burnside Basis Theorem that if G is a finite p-group with d(G)=r then given any generating set A for G there exists a subset of A of size r that generates G. We have denoted this property B. A group is said to have the basis property if all subgroups have property B. This thesis is a study into the nature of these two properties. Note all groups are finite unless stated otherwise. We begin this thesis by providing examples of groups with and without property B and several results on the structure of groups with property B, showing that under certain conditions property B is inherited by quotients. This culminates with a result which shows that groups with property B that can be expressed as direct products are exactly those arising from the Burnside Basis Theorem. We also seek to create a class of groups which have property B. We provide a method for constructing groups with property B and trivial Frattini subgroup using finite fields. We then classify all groups G where the quotient of G by the Frattini subgroup is isomorphic to this construction. We finally note that groups arising from this construction do not in general have the basis property. Finally we look at groups with the basis property. We prove that groups with the basis property are soluble and consist only of elements of prime-power order. We then exploit the classification of all such groups by Higman to provide a complete classification of groups with the basis property.

The purpose of this thesis was to present the technique of double coset enumeration and apply it to construct finite homomorphic images of infinite semidirect products. Several important homomorphic images include the classical groups, the Projective Special Linear group and the Derived Chevalley group were constructed.

The purpose of the thesis is to give an alternative and more efficient method for working with finite groups by constructing finite groups as homomorphic images of progenitors. The method introduced can be applied to all finite groups that possess symmetric generating sets of involutions. Such groups include all finite non-abelian simple groups, which can then be constructed by the technique of manual double coset enumeration.

Advantages of the double coset enumeration technique include its use to represent group elements in a convenient shorter form than their usual permutation representations and to find nice permutation representations for groups. In this thesis we construct, by hand, several groups, including U₃(3) : 2, L₂(13), PGL₂(11), and PGL₂(7), represent their elements in the short form (symmetric representation) and produce their permutation representations.

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: See the full text for the abstract / AFRIKAANSE OPSOMMING: Sien die volteks vir die opsomming

This paper presents a theory of Fourier transforms of complex-valued functions on a finite abelian group and investigates two applications of this theory. Chapter I is an introduction with remarks on notation. Basic theory, including Pontrvagin duality and the Poisson Summation formula, is the subject of Chapter II. In Chapter III the Fourier transform is viewed as an intertwining operator for certain unitary group representations. The solution of the eigenvalue problem of the Fourier transform of functions on the group Z/n of integers module n leads to a proof of the quadratic reciprocity law in Chapter IV. Chapter V addresses the, use of the Fourier transform in computing.

Suppose that G is a finite, unitary reflection group acting on a complex vector space V and X is a subspace of V. Define N to be the setwise stabilizer of X in G, Z to be the pointwise stabilizer, and C=N/Z. Then restriction defines a homomorphism from the algebra of G-invariant polynomial functions on V to the algebra of C-invariant functions on X. In my thesis, I extend earlier work by Douglass and Röhrle for Coxeter groups to the case where G is a complex reflection group of type G(r,p,n) in the notation of Shephard and Todd and X is in the lattice of the reflection arrangement of G. The main result characterizes when the restriction mapping is surjective in terms of the exponents of G and C and their reflection arrangements.

A logarithmic signature (LS) for a nite group G is an ordered tuple = [A1;A2; : : : ;An] of subsets Ai of G, such that every element g 2 G can be expressed uniquely as a product g = a1a2 : : : ; an, where ai 2 Ai. Logarithmic signatures were dened by Magliveras in the late 1970's for arbitrary nite groups in the context of cryptography. They were also studied for abelian groups by Hajos in the 1930's. The length of an LS is defined to be `() = Pn i=1 jAij. It can be easily seen that for a group G of order Qk j=1 pj mj , the length of any LS for G satises `() Pk j=1mjpj . An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS). The MLS conjecture states that every finite simple group has an MLS. If the conjecture is true then every finite group will have an MLS. The conjecture was shown to be true by a number of researchers for a few classes of finite simple groups. However, the problem is still wide open. This dissertation addresses the MLS conjecture for the classical simple groups. In particular, it is shown that MLS's exist for the symplectic groups Sp2n(q), the orthogonal groups O 2n(q0) and the corresponding simple groups PSp2n(q) and 2n(q0) for all n 2 N, prime power q and even prime power q0. The existence of an MLS is also shown for all unitary groups GUn(q) for all odd n and q = 2s under the assumption that an MLS exists for GUn 1(q). The methods used are very general and algorithmic in nature and may be useful for studying all nite simple groups of Lie type and possibly also the sporadic groups. The blocks of logarithmic signatures constructed in this dissertation have cyclic structure and provide a sort of cyclic decomposition for these classical groups. / by Nikhil Singhi. / Thesis (Ph.D.)--Florida Atlantic University, 2011. / Includes bibliography. / Electronic reproduction. Boca Raton, Fla., 2011. Mode of access: World Wide Web.