The study of formal power series is an area of interest that spans many areas of mathematics. We begin by looking at single-variable formal power series with coefficients from a field. By restricting to those series which are invertible with respect to formal composition we form a group. Our focus on this group focuses on the classification of elements having finite order. The notion of a semi-cyclic group comes up in this context, leading to several interesting results about torsion subgroups of the group. We then expand our focus to the composition of multivariate formal power series, looking at similar questions about classifying elements of finite order. We end by defining a natural automorphism on this group induced by a group action of the symmetric group.

In 1935, Philip Hall developed a formula for finding the number of ways of generating the group of symmetries of the icosahedron from a given number of its elements. In doing so, he defined a generalized Eulerian function. This thesis uses Hall's generalized Eulerian function to calculate generalized Eulerian functions for specific groups, namely: cyclic groups, dihedral groups, and p- groups.

When choosing k random elements from a group the kth expectation number is the expected size of the subgroup generated by those specific elements. The main purpose of this thesis is to study the asymptotic properties for the first and second expectation numbers of large cyclic groups. The first chapter introduces the kth expectation number. This formula allows us to determine the expected size of any group. Explicit examples and computations of the first and second expectation number are given in the second chapter. Here we show example of both cyclic and dihedral groups. In chapter three we discuss arithmetic functions which are crucial to computing the first and second expectation numbers. The fourth chapter is where we introduce and prove asymptotic results for the first expectation number of large cyclic groups. The asymptotic results for the second expectation number of cyclic groups is given in the fifth chapter. Finally, the results are summarized and future work for expectation numbers is discussed.

The Hochschild cohomology of an associative algebra records information about the deformations of that algebra, and hence the first step toward understanding its deformations is an examination of the Hochschild cohomology. In this dissertation, we use techniques from homological algebra, invariant theory, and combinatorics to analyze the Hochschild cohomology of skew group algebras arising from finite cyclic groups acting on polynomial rings over fields of arbitrary characteristic. These algebras are the natural semidirect product of the group ring with the polynomial ring. Many families of algebras arise as deformations of skew group algebras, such as symplectic reflection algebras and rational Cherednik algebras. We give an explicit description of the Hochschild cohomology governing graded deformations of skew group algebras for cyclic groups acting on polynomial rings. For skew group algebras, a description of the Hochschild cohomology is known in the nonmodular setting (i.e., when the characteristic of the field and the order of the group are coprime). However, in the modular setting (i.e., when the characteristic of the field divides the order of the group), much less is known, as techniques commonly used in the nonmodular setting are not available.

Neste trabalho inicialmente estudamos o rank da co-homologia do espaço dos pontos fixos de uma \'Z IND.p\' - ação semilivre sobre espaços X~p \' S POT. n\' x \'S POT.n\' e X~p \'S POT.n\' x \'S POT.n\' x \'S POT.n\' , com n>0. Em seguida, estudamos uma extensão para ações de p-grupos sobre espaços X~p \'S POT.n\' X \'S POT.m\', com 0< n \'< OU =\' m. Como parte do material utilizado demos uma descrição do diferencial d1 de uma seqüência espectral que converge para co-homologia equivariante de Tate, bem como uma versão da Fórmula de Künneth para a co-homologia equivariante de Tate. Na parte final, motivado pelo problemas de descrição de espaços de órbita de ações de grupos infinito, calculamos as co-homologias dos grupos virtualmente cíclicos (\'Z IND.a\' X| \' Z IND. b\' )X| Z e [\'Z POT.a\' X|(\'Z IND.b\' X \'Q IND. 2 POT.i\') ]X| Z / In this work is studied the rank of the fixed point set of a semifree action on spaces X~p \'S POT.n\' X \'S POT.n\' and X~p \'S POT.n\' X \'S POT.n\' X \'S POT.n\' , with n>0. We also consider the extension of the result for actions of p-groups on spaces X~p \'SPOT.n\' X \' S POT.m\' , with 0<n \'< OR =\' m. As result of the techniques used, we give a description of the differential d1 of a spectral sequence that converges to Tate equivariant cohomology, as well a version of the Künneth Formule to Tate equivariant cohomology. At the end, motivated by the space form problem for infinite groups we compute the cohomology of the virtually cyclic groups (\'Z IND. a\' X| \'Z IND. b\' )X| Z and [\'Z IND.a\' X|(\'Z IND. b\' X \'Q IND2 POT. i\' )] X| Z

Seja Cp um grupo cíclico de ordem p, onde p é um número primo tal que S = {1, , 1+\\theta, 1+\\theta+\\theta^2, · · · , 1 +\\theta + · · · + \\theta ^{p-3/2}} gera o grupo das unidades de Z[\\theta] e é uma raiz p-ésima primitiva da unidade sobre Q. No artigo \"Units of ZCp\" , Ferraz apresentou um modo simples de encontrar um conjunto de geradores independentes para o grupo das unidades do anel de grupo ZCp sobre os inteiros. Nós estendemos este resultado para ZCp^n , considerando que um conjunto similar a S gera o grupo das unidades de Z[\\theta]. Isto ocorre, por exemplo, quando \\phi(p^n)\\leq 66. Descrevemos o grupo das unidades de ZCp^n como o produto ±ker(\\pi_1) × Im(\\pi1), onde \\pi_1 é um homomorfismo de grupos. Além disso, explicitamos as bases de ker(\\pi_1) e Im(\\pi_1). / Let Cp be a cyclic group of order p, where p is a prime integer such that S = {1, , 1 + \\theta, 1 +\\theta +\\theta ^2 , · · · , 1 + \\theta + · · · +\\theta ^{p-3/2}} generates the group of units of Z[\\theta] and is a primitive pth root of 1 over Q. In the article \"Units of ZCp\" , Ferraz gave an easy way to nd a set of multiplicatively independent generators of the group of units of the integral group ring ZCp . We extended this result for ZCp^n , provided that a set similar to S generates the group of units of Z[\\theta]. This occurs, for example, when \\phi(p^n)\\leq 66. We described the group of units of ZCp^n as the product ±ker(\\pi_1) × Im(\\pi_1), where \\pi_1 is a group homomorphism. Moreover, we explicited a basis of ker(\\pi_1) and I m(\\pi_1).

Submitted by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-05-19T15:20:50Z No. of bitstreams: 2 arquivototal.pdf: 1103922 bytes, checksum: fee5e8830b60905917fc3ab1fb8c2aae (MD5) license_rdf: 22190 bytes, checksum: 19e8a2b57ef43c09f4d7071d2153c97d (MD5) / Approved for entry into archive by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-05-19T15:21:56Z (GMT) No. of bitstreams: 2 arquivototal.pdf: 1103922 bytes, checksum: fee5e8830b60905917fc3ab1fb8c2aae (MD5) license_rdf: 22190 bytes, checksum: 19e8a2b57ef43c09f4d7071d2153c97d (MD5) / Made available in DSpace on 2015-05-19T15:21:56Z (GMT). No. of bitstreams: 2 arquivototal.pdf: 1103922 bytes, checksum: fee5e8830b60905917fc3ab1fb8c2aae (MD5) license_rdf: 22190 bytes, checksum: 19e8a2b57ef43c09f4d7071d2153c97d (MD5) Previous issue date: 2013-08-13 / The encryption algorithm written by Egyptian Taher ElGamal computes discrete logarithms with elements of a finite group G Cyclical. These elements have properties that during the study Chapter 1. Knowing the definitions and some properties studied, we can define and compute discrete logarithms, using knowledge of arithmetic and congruence of Remains and Theorem Remainder of Chinese. We will study public key algorithms, in particular the algorithm written by ElGamal, seeking to understand the diffculties presented by it and show its applications in the field of cryptography. We present a sequence of activities, aimed at students of the first grade of high school, targeting the learning of some subjects covered at work. / O algoritmo de criptografia escrito pelo egípcio Taher ElGamal calcula logaritmos discretos com elementos de um Grupo Cíclico finito G. Esses elementos possuem propriedades que estudaremos no decorrer do capítulo 1. Conhecendo as definições e algumas propriedades estudadas, poderemos definir e calcular logaritmos discretos, utilizando conhecimentos da Aritmética dos Restos e Congruências, bem como o Teorema Chinês dos Restos. Vamos estudar algoritmos de chave pública, em particular o algoritmo escrito por ElGamal, buscando entender as dificuldades apresentadas por ele e mostrar suas aplicações no campo da Criptografia. Apresentaremos uma sequencia de atividades, voltadas para estudantes do primeiro ano do Ensino Médio, visando o aprendizado de alguns assuntos abordados no trabalho.

Neste trabalho inicialmente estudamos o rank da co-homologia do espaço dos pontos fixos de uma \'Z IND.p\' - ação semilivre sobre espaços X~p \' S POT. n\' x \'S POT.n\' e X~p \'S POT.n\' x \'S POT.n\' x \'S POT.n\' , com n>0. Em seguida, estudamos uma extensão para ações de p-grupos sobre espaços X~p \'S POT.n\' X \'S POT.m\', com 0< n \'< OU =\' m. Como parte do material utilizado demos uma descrição do diferencial d1 de uma seqüência espectral que converge para co-homologia equivariante de Tate, bem como uma versão da Fórmula de Künneth para a co-homologia equivariante de Tate. Na parte final, motivado pelo problemas de descrição de espaços de órbita de ações de grupos infinito, calculamos as co-homologias dos grupos virtualmente cíclicos (\'Z IND.a\' X| \' Z IND. b\' )X| Z e [\'Z POT.a\' X|(\'Z IND.b\' X \'Q IND. 2 POT.i\') ]X| Z / In this work is studied the rank of the fixed point set of a semifree action on spaces X~p \'S POT.n\' X \'S POT.n\' and X~p \'S POT.n\' X \'S POT.n\' X \'S POT.n\' , with n>0. We also consider the extension of the result for actions of p-groups on spaces X~p \'SPOT.n\' X \' S POT.m\' , with 0<n \'< OR =\' m. As result of the techniques used, we give a description of the differential d1 of a spectral sequence that converges to Tate equivariant cohomology, as well a version of the Künneth Formule to Tate equivariant cohomology. At the end, motivated by the space form problem for infinite groups we compute the cohomology of the virtually cyclic groups (\'Z IND. a\' X| \'Z IND. b\' )X| Z and [\'Z IND.a\' X|(\'Z IND. b\' X \'Q IND2 POT. i\' )] X| Z

Seja Cp um grupo cíclico de ordem p, onde p é um número primo tal que S = {1, , 1+\\theta, 1+\\theta+\\theta^2, · · · , 1 +\\theta + · · · + \\theta ^{p-3/2}} gera o grupo das unidades de Z[\\theta] e é uma raiz p-ésima primitiva da unidade sobre Q. No artigo \"Units of ZCp\" , Ferraz apresentou um modo simples de encontrar um conjunto de geradores independentes para o grupo das unidades do anel de grupo ZCp sobre os inteiros. Nós estendemos este resultado para ZCp^n , considerando que um conjunto similar a S gera o grupo das unidades de Z[\\theta]. Isto ocorre, por exemplo, quando \\phi(p^n)\\leq 66. Descrevemos o grupo das unidades de ZCp^n como o produto ±ker(\\pi_1) × Im(\\pi1), onde \\pi_1 é um homomorfismo de grupos. Além disso, explicitamos as bases de ker(\\pi_1) e Im(\\pi_1). / Let Cp be a cyclic group of order p, where p is a prime integer such that S = {1, , 1 + \\theta, 1 +\\theta +\\theta ^2 , · · · , 1 + \\theta + · · · +\\theta ^{p-3/2}} generates the group of units of Z[\\theta] and is a primitive pth root of 1 over Q. In the article \"Units of ZCp\" , Ferraz gave an easy way to nd a set of multiplicatively independent generators of the group of units of the integral group ring ZCp . We extended this result for ZCp^n , provided that a set similar to S generates the group of units of Z[\\theta]. This occurs, for example, when \\phi(p^n)\\leq 66. We described the group of units of ZCp^n as the product ±ker(\\pi_1) × Im(\\pi_1), where \\pi_1 is a group homomorphism. Moreover, we explicited a basis of ker(\\pi_1) and I m(\\pi_1).

Dado um grupo G, a definição dos grupos de cohomologia com coeficientes em um ZG-módulo M podem ser dadas usando as técnicas usuais da Álgebra Homológica, que garantem a existência de resoluções projetivas P de Z como um ZG-módulo trivial, a equivalência entre resoluções distintas etc. Podemos também construir o produto cup em cohomologia, cuja definição depende de uma aproximação da diagonal para a resolução projetiva P. Entretanto, o cálculo explicito de tais resoluções e dos grupos de cohomologia pode ser bastante difícil na prática, e ainda mais difícil a obtenção de uma aproximação da diagonal. Nesta tese, obteremos resoluções livres e aproximações da diagonal para os grupos fundamentais das superfícies que são espaços K(G,1) e também para o grupo fundamental de fibrados do toro com base S^1, bem como a estrutura de anel de cohomologia de tais grupos. Ainda, para certos grupos virtualmente cíclicos G, obteremos o anel de cohomologia calculando diretamente uma resolução livre e uma aproximação da diagonal, ou então usando a sequência espectral de Lyndon-Hochschild-Serre. A motivação para o estudo da primeira família de grupos vem do fato de representarem variedades de dimensão 2 e 3, e da segunda família por ser constituída de grupos que atuam em esferas de homotopia. / Given a group G, a definition for its cohomology groups with coefficients in a given ZG-module M can be given using the standard techniques of Homological Algebra, that ensure the existence of projective resolutions P of Z as a trivial ZG-module, the equivalence between two such resolutions etc . We can also construct the cup product, whose definition depends on a diagonal approximation for a given projective resolution P. However, the explicit computation of such resolutions and of the cohomology groups may be very hard in practice, and even worse may be the task of constructing a diagonal approximation. In this thesis, we obtain free resolutions and diagonal approximations for the fundamental groups of surfaces that are K(G,1) spaces and for the fundamental group of the torus bundle with the circle as the base space, as well as the structure of the cohomology ring of these groups. Also, for some virtually cyclic groups, we obtain the cohomology ring by an explicit computation of a free resolution and a diagonal approximation, or by the Lyndon-Hochschild-Serre spectral sequence. The motivation for the study of the first family of groups comes from the fact that such groups represent manifolds of dimension 2 and 3, and the groups of the second family act on homotopy spheres.