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 (Ph. D.)--University of Oregon, 2007. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 107-108). Also available for download via the World Wide Web; free to University of Oregon users.

Orientador: Claudemir Murari / Banca: Marcelo José Saia / Banca: Miriam Godoy Penteado / Resumo: Este trabalho teve o objetivo de produzir um conjunto de atividades para analisar como o uso do caleidoscópio associado ao estudo dos ornamentos planos pode contribuir no ensino de grupos de simetria e transformações geométricas em um curso de graduação em Matemática. Esta pesquisa tem caráter qualitativo e foi desenvolvida segundo a proposta metodológica de Romberg. Elaborou-se uma proposta de ensino baseada na metodologia de Resolução de Problemas que foi aplicada a um grupo de professores (alguns em fase de formação) de matemática. As atividades tiveram a finalidade de fazer com que os alunos usassem o caleidoscópio para reproduzir ornamentos planos e, a partir de então, discutissem, com base em argumentos geométricos e algébricos, quais as possibilidades (e impossibilidades) que esse instrumento oferece para obtenção desses ornamentos e suas respectivas justificativas. A coleta de dados ocorreu, essencialmente, por observação participante em sala de aula por meio do uso de questionários, anotações e registros fotográficos. Após a coleta de dados, foi feita uma análise das possibilidades e limitações do material desenvolvido para o ensino de grupos de simetria e transformações geométricas, bem como o uso do caleidoscópio enquanto recurso didático / Abstract: The purpose of this work was to develop a set of activities to analyze how the use of kaleidoscope associated to the study of ornaments can contribute to the teaching of symmetry groups and geometric transformations on a undergraduate course in Mathematics. This is a qualitative research and it was developed according to the methodological proposal of Romberg. A teaching proposal was drafted and was applied to a group of mathematics teachers. Activities were designed following the methodology of problem-solving and intended to make students to use the kaleidoscope to reproduce some ornaments and thereafter, discuss, based on geometric and algebraic arguments, the possibilities and impossibilities that this tool provides to obtain ornaments and their respective justifications. Data collection occurred primarily by participant observation in the classroom through the use of questionnaires, notes and photographic records. After the end of the course a viability analysis of the activities was done (possibilities and limitations) for teaching symmetry groups and geometric transformations as well as the use of Kaleidoscope as a didactic tool / Mestre

This thesis presents and evaluates a new parallel algorithm that computes all single cosets in the double coset M₂₄ P M₂₄, where P is a permutation on n points of a certain cycle structure, and M₂₄ is the Mathieu group related to a Steiner system S(5, 8, 24) as its automorphism group. The purpose of this work is not to replace the existing algorithms, but rather to explore a possibility to extend calculations of single cosets beyond the limits encountered when using currently available methods.

Orientador : Sueli Irene Rodrigues Costa / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-06T14:39:59Z (GMT). No. of bitstreams: 1 Siqueira_RogerioMonteirode_D.pdf: 1994309 bytes, checksum: 7735d63966bc2d9b5c84ccac989c3289 (MD5) Previous issue date: 2006 / Resumo: Códigos esféricos euclidianos com simetrias são órbitas finitas de grupos de matrizes ortogonais. Tais códigos são também conhecidos como códigos de grupo. Neste trabalho, os códigos de grupo comutativo em dimensão par são caracterizados sobre toros planos, subvariedades da esfera. Em particular, se o grupo de matrizes for cíclico, o código gerado está contido em um nó que se enrola em um tora. Se a dimensão for ímpar, todo código de grupo comutativo mora em anti-primas cujas bases estão contidas em dois toros planos. Tal caracterização permitiu a construção de limitantes para a cardinalidade destas constelações de pontos em termos da distância mínima destes códigos e da densidade de empacotamento de um reticulado associado. Utilizando o método de Biglieri e Elia, que procura o vetor inicial cujo respectivo código de grupo cíclico tem a melhor distância mínima, apresentamos também os melhores códigos de grupo cíclico em dimensão quatro até 100 pontos / Abstract: Euclidean spherical codes with symmetries are orbits of finite orthogonal matrix groups. These codes are also known as group codes. ln this work, the commutative group codes in even dimensions are viewed on flat tori, which are submanifolds of the sphere. Also, if the matrix group is cyclic, the generated code lies on a knot which wraps around a torus. If the dimension is odd, every commutative group code lies on an anti-prism whose bases are contained in two flat tori. This interpretation lead us to build upper bounds for the cardinality of these constellations involving their minimum distance and the packing density of an associated lattice. Using a method by Biglieri and Elia, which searchs the initial vector for a cyclic group in order to achieve the best minimum distance, we also present the best cyclic group codes in dimension four up to 100 points / Doutorado / Matematica / Doutor em Matemática

Orientador: Claudina Izepe Rodrigues / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-27T16:14:16Z (GMT). No. of bitstreams: 1 Franco_MarciaCristinaLemosGuimaraes_M.pdf: 20497304 bytes, checksum: c28b5c2e4a775ec0c3f67197069a584f (MD5) Previous issue date: 2015 / Resumo: Neste trabalho apresentamos um estudo sobre grupos, transformações geométricas e isometrias no plano. Apresentamos o teorema da classificação das isometrias no plano, o teorema de Leonardo que classifica os grupos de simetria de ornamentos limitados e o teorema da classificação dos grupos de frisos. Propomos sequências de atividades para a Educação Básica envolvendo as isometrias e a identificação do grupo de simetria de um ornamento limitado e de um friso. Além disso, as atividades sugeridas apresentam intuitivamente a ideia da estrutura algébrica de grupos. Finalizamos este trabalho relatando como ocorreu a aplicação de três das sequências sugeridas, os procedimentos adotados e os resultados obtidos / Abstract: We present a study of groups, geometric transformations and isometries in the plane. Introducing the classification theorem of isometries in the plane, the Leonardo theorem that classifies symmetry groups of limited ornaments and the classification theorem of friezes groups. We propose a sequence of activities for the basic education involving isometry and identification of symmetry group of limited ornaments and friezes. Moreover some of the suggested activities provide an intuitive idea of the algebraic structure of groups. We end this paper by reporting on the manner in which the application of three of the suggested sequences occurred, the procedures that were adopted and the results that were obtained / Mestrado / Matemática em Rede Nacional / Mestra

In Chapters 1 and 2, we introduce the definition of interpolating refinable function vectors in dimension one and high dimensions, characterize such interpolating refinable function vectors in terms of their masks, and derive their sum rule structure explicitly. We study biorthogonal refinable function vectors from interpolating refinable function vectors. We also study the symmetry property of an interpolating refinable function vector and characterize a symmetric interpolating refinable function vector in any dimension with respect to certain symmetry group in terms of its mask. Examples of interpolating refinable function vectors with some desirable properties, such as orthogonality, symmetry, compact support, and so on, are constructed according to our characterization results. In Chapters 3 and 4, we turn to the study of general matrix extension problems with symmetry for the construction of orthogonal and biorthogonal multiwavelets. We give characterization theorems and develop step-by-step algorithms for matrix extension with symmetry. To illustrate our results, we apply our algorithms to several examples of interpolating refinable function vectors with orthogonality or biorthogonality obtained in Chapter 1. In Chapter 5, we discuss some possible future research topics on the subjects of matrix extension with symmetry in high dimensions and frequency-based non-stationary tight wavelet frames with directionality. We demonstrate that one can construct a frequency-based tight wavelet frame with symmetry and show that directional analysis can be easily achieved under the framework of tight wavelet frames. Potential applications and research directions of such tight wavelet frames with directionality are discussed. / Applied Mathematics

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