The goal of this paper is to explore the first cohomology group of groups G that are not necessarily finitely generated. Our focus is on l^p-cohomology, 1 leq p leq infty, and what results regarding finitely generated groups change when G is infinitely generated. In particular, for abelian groups and locally finite groups, the l^p-cohomology is non-zero when G is countable, but vanishes when G has sufficient cardinality. We then show that the l^infty-cohomology remains unchanged for many classes of groups, before looking at several results regarding the injectivity of induced maps from embeddings of G-modules. We present several new results for countable groups, and discuss which results fail to hold in the general uncountable case. Lastly, we present results regarding reduced cohomology, including a useful lemma extending vanishing results for finitely generated groups to the infinitely generated case. / Ph. D. / The goal of this paper is to use a technique that originated in algebraic topology to study the properties of a structure called a group. Groups are collections of objects that interact with each other through an operation that obeys certain properties. Groups arise when considering many different mathematical questions, and they were first studied when looking at the different symmetries an object can have. Classifying the different properties of a group is an active area of mathematical research. We seek to do this by looking at collections of maps from a particular group to the real or complex numbers, then studying how the group shifts these functions.

We want to take a look at the first cohomology group H^1(G, l^2(G)), in particular when G is locally-finite. First, though, we discuss some results about the space H^1(G, C G) for G locally-finite, as well as the space H^1(G, l^2(G)) when G is finitely generated. We show that, although in the case when G is finitely generated the embedding of C G into l^2(G) induces an embedding of the cohomology groups H^1(G, C G) into H^1(G, l^2(G)), when G is countably-infinite locally-finite, the induced homomorphism is not an embedding. However, even though the induced homomorphism is not an embedding, we still have that H^1(G, l^2(G)) neq 0 when G is countably-infinite locally-finite. Finally, we give some sufficient conditions for H^1(G,l^2(G)) to be zero or non-zero. / Master of Science

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A group G is called locally finite if every finitely generated subgroup of G is finite. In this thesis we study the centralizers of subgroups in simple locally finite groups. Hartley proved that in a linear simple locally finite group, the fixed point of every semisimple automorphism contains infinitely many elements of distinct prime orders. In the first part of this thesis, centralizers of finite abelian subgroups of linear simple locally finite groups are studied and the following result is proved: If G is a linear simple locally finite group and A is a finite d-abelian subgroup consisting of semisimple elements of G, then C_G(A) has an infinite abelian subgroup isomorphic to the direct product of cyclic groups of order p_i for infinitely many distinct primes pi. Hartley asked the following question: Let G be a non-linear simple locally finite group and F be any subgroup of G. Is CG(F) necessarily infinite? In the second part of this thesis, the following problem is studied: Determine the nonlinear simple locally finite groups G and their finite subgroups F such that C_G(F) contains an infinite abelian subgroup which is isomorphic to the direct product of cyclic groups of order pi for infinitely many distinct primes p_i. We prove the following: Let G be a non-linear simple locally finite group with a split Kegel cover K and F be any finite subgroup consisting of K-semisimple elements of G. Then the centralizer C_G(F) contains an infinite abelian subgroup isomorphic to the direct product of cyclic groups of order p_i for infinitely many distinct primes p_i.

In any group G, we may extend the definition of the conjugacy class of an element to the conjugacy class of a k-tuple, for a positive integer k. When k = 2, we are forming the conjugacy classes of ordered pairs, when k = 3, we are forming the conjugacy classes of ordered triples, etc. In this report we explore a generalized question which Professor B. Doug Park has posed (for k = 2). For an arbitrary k, is it true that: (G has finitely many k-conjugacy classes) implies (G is finite)? Supposing to the contrary that there exists an infinite group G which has finitely many k-conjugacy classes for all k = 1, 2, 3, ..., we present some preliminary analysis of the properties that G must have. We then investigate known classes of groups having some of these properties: universal locally finite groups, existentially closed groups, and Engel groups.

In any group G, we may extend the definition of the conjugacy class of an element to the conjugacy class of a k-tuple, for a positive integer k. When k = 2, we are forming the conjugacy classes of ordered pairs, when k = 3, we are forming the conjugacy classes of ordered triples, etc. In this report we explore a generalized question which Professor B. Doug Park has posed (for k = 2). For an arbitrary k, is it true that: (G has finitely many k-conjugacy classes) implies (G is finite)? Supposing to the contrary that there exists an infinite group G which has finitely many k-conjugacy classes for all k = 1, 2, 3, ..., we present some preliminary analysis of the properties that G must have. We then investigate known classes of groups having some of these properties: universal locally finite groups, existentially closed groups, and Engel groups.

abstract INERT SUBGROUPS AND CENTRALIZERS OF INVOLUTIONS IN LOCALLY FINITE SIMPLE GROUPS &uml / Ozyurt, Erdal Ph. D., Department of Mathematics Supervisor: Prof. Dr. Mahmut Kuzucuo&amp / #728 / glu September 2003, 68 pages A subgroup H of a group G is called inert if [H : H Hg] is finite for all g 2 G. A group is called totally inert if every subgroup is inert. Among the basic properties of inert subgroups, we prove the following. Let M be a maximal subgroup of a locally finite group G. If M is inert and abelian, then G is soluble with derived length at most 3. In particular, the given properties impose a strong restriction on the derived length of G. We also prove that, if the centralizer of every involution is inert in an infinite locally finite simple group G, then every finite set of elements of G can not be contained in a finite simple group. In a special case, this generalizes a Theorem of Belyaev&amp / #8211 / Kuzucuo&amp / #728 / glu&amp / #8211 / Se&cedil / ckin, which proves that there exists no infinite locally finite totally inert simple group.

For $n\geq2$, a variety $\mathcal{V}$ is said to be congruence $n$-permutable if every algebra $\mathbf{A}\in\mathcal{V}$ satisfies $\alpha\circ^n\beta=\beta\circ^n\alpha$, for all $\alpha,\beta\in \Con(\mathbf{A})$. Furthermore, given any algebra $\mathbf{A}$ and $k\geq1$, a $k$-dimensional Hagemann relation on $\mathbf{A}$ is a reflexive compatible relation $R\subseteq A\times A$ such that $R^{-1}\not\subseteq R\circ^k R$. A famous result of J. Hagemann and A. Mitschke shows that a variety $\mathcal{V}$ is congruence $n$-permutable if and only if $\mathcal{V}$ has no member carrying an $(n-1)$-dimensional Hagemann relation: by using this criterion, we provide another Maltsev characterization of congruence $n$-permutability, equivalent to the well-known Schmidt's and Hagemann-Mitschke's (\cite{HagMit}) term-based descriptions. We further establish that the omission by varieties of certain special configurations of Hagemann relations induces the satisfaction of suitable Maltsev conditions. These omission properties may be used to characterize congruence $n$-permutable idempotent varieties for some $n\geq2$, congruence 2-permutable idempotent varieties and congruence 3-permutable locally finite idempotent varieties, yielding that the following are prime Maltsev conditions: \begin{enumerate} \item congruence $n$-permutability for some $n\geq2$ with respect to idempotent varieties; \item congruence 2-permutability with respect to idempotent varieties; \item congruence 3-permutability with respect to locally finite idempotent varieties. \end{enumerate} Finally, we focus on the analysis of a family of strong Maltsev conditions, which we denote by $\{\mathcal{D}_n:2\leq n<\omega\}$, such that any variety $\mathcal{V}$ is congruence $n$-permutable whenever $\mathcal{D}_n$ is interpretable in $\mathcal{V}$. Among various other properties, we also show that the $\mathcal{D}_n$'s with odd $n\geq3$ generate decomposable strong Maltsev filters in the lattice of interpretability types. / Thesis / Doctor of Philosophy (PhD)

This work was motivated by the famous Plateau\'s Problem which concerns the existence of a minimizing set of the area functional with prescribed boundary. In order to solve the Plateau\'s Problem, we make use of different theories: the theory of varifolds, currents and locally finite perimeter sets (Caccioppoli sets). Working on the Caccioppoli sets theory, it is straightforward to prove the existence of a minimizing set in some classical problems as the isoperimetric and Plateau\'s problems. If we switch the problem to find the regularity that we can extract of some minimizing set, we come across complicated ideas and tools. Although, the Plateau\'s Problem and other classical problems are well settled. Because of that, we have extensively studied the almost minimizing condition ((; r)-minimizing sets) considered by Maggi ([?]) which subsumes some classical problems. We focused on the regularity theory extracted from this almost minimizing condition. / This work was motivated by the famous Plateau\'s Problem which concerns the existence of a minimizing set of the area functional with prescribed boundary. In order to solve the Plateau\'s Problem, we make use of different theories: the theory of varifolds, currents and locally finite perimeter sets (Caccioppoli sets). Working on the Caccioppoli sets theory, it is straightforward to prove the existence of a minimizing set in some classical problems as the isoperimetric and Plateau\'s problems. If we switch the problem to find the regularity that we can extract of some minimizing set, we come across complicated ideas and tools. Although, the Plateau\'s Problem and other classical problems are well settled. Because of that, we have extensively studied the almost minimizing condition ((; r)-minimizing sets) considered by Maggi ([?]) which subsumes some classical problems. We focused on the regularity theory extracted from this almost minimizing condition.

Made available in DSpace on 2014-07-29T16:02:19Z (GMT). No. of bitstreams: 1 Dissertacao Marcio Lima.pdf: 1529346 bytes, checksum: c6a80a13d55b40203c44877c4cdeb1f4 (MD5) Previous issue date: 2011-06-27 / A be an elementary abelian group of order q2, where q a prime number. In this paper we will study the influence of centering on the structure of automorphism groups profinitos in this sense if A acting as a coprime group of automorphisms on a group profinito G and CG(a) is periodic for each a 2 A#, then we will show that G is locally finite. It will be demonstrated also the case where A acts as a group of automorphisms of a group pro-p of G / Sejam A um grupo abeliano elementar de ordem q2, onde q um número primo. Neste trabalho estudamos a influência dos centralizadores de automorfismos na estrutura dos grupos profinitos, neste sentido se A age como um grupo de automorfismos coprimos sobre um grupo profinito G e que CG(a) é periódico para cada a 2 A#, então mostraremos que G é localmente finito. Será demonstrado também o caso onde A age como um grupo de automorfismos sobre um grupo pro-p de G.