Thesis (M.A.)--State University of New York at Binghamton, Department of Mathematical Sciences, 2009. / Includes bibliographical references.

In the maximum solution equation problem a collection of equations are given over some algebraic structure. The objective is to find an assignment to the variables in the equations such that all equations are satisfied and the sum of the variables is maximised. We give tight approximability results for the maximum solution equation problem when the equations are given over finite abelian groups. We also prove that the weighted and unweighted versions of this problem have asymptotically equal approximability thresholds. Furthermore, we show that the problem is equally hard to solve as the general problem even if each equation is restricted to contain at most three variables and solvable in polynomial time if the equations are restricted to contain at most two variables each. All of our results also hold for the generalised version of maximum solution equation where the elements of the group are mapped arbitrarily to non-negative integers in the objective function.

Computational group theory deals with the design, analysis and computer implementation of algorithms for solving computational problems involving groups, and with the applications of the programs produced to interesting questions in group theory, in other branches of mathematics, and in other areas of science. This thesis describes an implementation of a proposal for a Soluble Quotient Algorithm, i.e. a description of the algorithms used and a report on the findings of an empirical study of the behaviour of the programs, and gives an account of an application of the programs. The programs were used for the construction of soluble groups with interesting properties, e.g. for the construction of soluble groups of large derived length which seem to be candidates for groups having efficient presentations. New finite soluble groups of derived length six with trivial Schur multiplier and efficient presentations are described. The methods for finding efficient presentations proved to be only practicable for groups of moderate order. Therefore, for a given derived length soluble groups of small order are of interest. The minimal soluble groups of derived length less than or equal to six are classified.

Neste trabalho resolvemos o problema da classicação dos possíveis esquemas de quebra de simetria que reproduzem as degenerescências do código genético na categoria dos grupos finitos simples, contribuindo assim para a busca de modelos algébricos para a evolução do código genético, iniciada por Hornos & Hornos. / In this work we solve the problem of classifying the possible symmetry breaking schemes based on simple finite groups that reproduce the degeneracies of the genetic code, thus contributing to the search for algebraic models that describe the evolution of the genetic code, initiated by Hornos & Hornos.

O objetivo principal deste trabalho é apresentar um estudo sobre o Teorema de Borsuk-Ulam para forma espacial esférica homotópica. Em nosso trabalho consideramos X uma n-forma espacial esférica homotópica a qual admite uma ação livre de Zp, com p> 2 primo e f : X → Rk uma função contínua e, mostramos que sob determinada relação entre os números n e k, o conjunto A(f) dos pontos de coincidência de f é não vazio / The main objective of this work is to present a study about the Borsuk- Ulam Theorem for homotopic spherical space. In our work we consider X be a n-dimensional homotopic spherical space form which admits a free action of Zp, with p> 2 prime and f : X → Rk be a continuous map and we show that, under certain relations between the numbers n and k, the set A(f) is not empty

Orientador: Dessislava Hristova Kochloukova / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T09:13:05Z (GMT). No. of bitstreams: 1 Rego_YuriSantos_M.pdf: 1142010 bytes, checksum: 548d8ef6ff2800026c8cd65783b81a9f (MD5) Previous issue date: 2014 / Resumo: Neste trabalho estuda-se os principais resultados dados por J. Wilson no artigo "Finite Presentations of Pro-p Groups and Discrete Groups", relacionados à Desigualdade de Golod-¿afarevi? para uma ampla classe de grupos pro-p e abstratos infinitos. Apresentamos a teoria básica de grupos livres abstratos, levando à noção de apresentação de grupos, com foco em apresentações finitas. É feito um estudo sobre grupos profinitos, particularmente no caso pro-p. Abrange-se definições, propriedades algébricas e topológicas básicas, bem como o caso de finitos geradores com o subgrupo de Frattini, e conceitos de completamentos, de grupos pro-p livres, de apresentações de grupos pro-p e de álgebras de grupo completas. No capítulo final estudamos os resultados principais para grupos pro-p e abstratos finitamente apresentáveis, que incluem grupos solúveis e implicações na estrutura de certos grupos satisfazendo a Desigualdade. Os anexos relacionam a teoria aqui apresentada a grupos pro-p de posto finito e homologia e cohomologia de grupos pro-p / Abstract: In this work we study the main results presented by J. Wilson in his paper "Finite Presentations of Pro-p Groups and Discrete Groups", which extend the Golod-¿afarevi? Inequality to a large class of infinite pro-p and abstract groups. In the first chapter we present the basic theory of abstract free groups, focusing on finite presentations. Next we study profinite groups, with focus on pro-p groups. This study ranges from definitions to basic algebraic and topological properties, as well as the cases of finitely generated groups and the Frattini subgroup, and notions of completion, free pro-p groups, presentations of pro-p groups and completed group algebras. In the last chapter we study the main results regarding finite presentations of pro-p and abstract groups, which include soluble groups and implications on the structure of certain groups for which the Inequality holds. In the appendixes we briefly relate the presented theory to pro-p groups of finite rank and homology and cohomology of pro-p groups / Mestrado / Matematica / Mestre em Matemática

Orientador: Plamen Emilov Kochloukov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-20T05:07:15Z (GMT). No. of bitstreams: 1 Mattos_AldaDayana_D.pdf: 2629516 bytes, checksum: d0038b12fc57a77bd28fa477bde8e99a (MD5) Previous issue date: 2012 / Resumo: Seja G um grupo finito que age fielmente sobre a álgebra das matrizes de ordem dois...Observação: O resumo, na íntegra, poderá ser visualizado no texto completo da tese digital / Abstract: Let G be a finite group acting faithfully on the matrix algebra of order two...Note: The complete abstract is available with the full electronic document / Doutorado / Matematica / Doutor em Matemática

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The purpose of this project was to study the classical result in this direction discovered by O. Ore in 1938, as well as related theorems and corollaries. Ore's Theorem and its corollaries provide us with several results relating distributive lattices with cyclic groups.

This project deals with the automorphism group G of a Steiner system S (3, 4, 10). S₁₀, the symmetrical group of degree 10, acts transitively on T, the set of all Steiner systems with parameters 3, 4, 10. The purpose of this project is to study the action of S₁₀ on cosets of G. This will be achieved by means of a graph of S₁₀ on T x T. The orbits of S₁₀ on T x T are in one-one correspondence with the orbits of G, the stabilizer of an S [e] T on T.