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2010 | Buch

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Profinite Groups

verfasst von: Luis Ribes, Pavel Zalesskii

Verlag: Springer Berlin Heidelberg

Buchreihe : A Series of Modern Surveys in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The aim of this book is to serve both as an introduction to profinite groups and as a reference for specialists in some areas of the theory. The book is reasonably self-contained. Profinite groups are Galois groups. As such they are of interest in algebraic number theory. Much of recent research on abstract infinite groups is related to profinite groups because residually finite groups are naturally embedded in a profinite group. In addition to basic facts about general profinite groups, the book emphasizes free constructions (particularly free profinite groups and the structure of their subgroups). Homology and cohomology is described with a minimum of prerequisites.

This second edition contains three new appendices dealing with a new characterization of free profinite groups, presentations of pro-p groups and a new conceptually simpler approach to the proof of some classical subgroup theorems. Throughout the text there are additions in the form of new results, improved proofs, typographical corrections, and an enlarged bibliography. The list of open questions has been updated; comments and references have been added about those previously open problems that have been solved after the first edition appeared.

Inhaltsverzeichnis

Frontmatter

1. Inverse and Direct Limits

Abstract

In this section we define the concept of inverse (or projective) limit and establish some of its elementary properties. Rather than developing the concept and establishing those properties under the most general conditions, we restrict ourselves to inverse limits of topological spaces or topological groups. We leave the reader the task of extending and translating the concepts and results obtained here to other objects such as sets, (topological) rings, modules, graphs…, or to more general categories.

Luis Ribes, Pavel Zalesskii

2. Profinite Groups

Abstract

Let $\mathcal{C}$ be a nonempty class of finite groups [this will always mean that $\mathcal{C}$ contains all the isomorphic images of the groups in $\mathcal{C}$]. Define a pro - $\mathcal{C}$ group G as an inverse limit

$$G=\lim\limits_{\displaystyle\longleftarrow\atop{i\in I}}G_i$$

of a surjective inverse system {G _i,φ _ij,I} of groups G _i in $\mathcal{C}$, where each group G _i is assumed to have the discrete topology. We think of such a pro - $\mathcal{C}$ group G as a topological group, whose topology is inherited from the product topology on ∏_i∈I G _i.

Luis Ribes, Pavel Zalesskii

3. Free Profinite Groups

Abstract

Let $\mathcal{N}$ be a nonempty collection of normal subgroups of finite index of a group G and assume that $\mathcal{N}$ is filtered from below, i.e., $\mathcal{N}$ satisfies the following condition:

$$\mbox{whenever }N_1,N_2\in \mathcal{N},\mbox{ there exists }N\in \mathcal{N}\mbox{ such that }N\le N_1\cap N_2.$$

Then one can make G into a topological group by considering $\mathcal{N}$ as a fundamental system of neighborhoods of the identity element 1 of G. We refer to the corresponding topology on G as a profinite topology. If every quotient G/N $(N\in \mathcal{N})$ belongs to a certain class $\mathcal{C}$, we say more specifically that the topology above is a pro - $\mathcal{C}$ topology.

Luis Ribes, Pavel Zalesskii

4. Some Special Profinite Groups

Abstract

Let G be a profinite group and x∈G. Since $\widehat {\mathbf{Z}}$ is a free profinite group on {1}, there is a unique epimorphism

$$\varphi :\widehat{\mathbf{Z}}\longrightarrow \overline {\langle x\rangle }$$

such that φ(1)=x. Given $\lambda \in \widehat {\mathbf{Z}}$, define x ^λ=φ(λ).

Luis Ribes, Pavel Zalesskii

5. Discrete and Profinite Modules

Abstract

A profinite ring Λ is an inverse limit of an inverse system {Λ _i,φ _ij} of finite rings. We always assume that rings have an identity element, denoted usually by 1, and that homomorphisms of rings send identity elements to identity elements. A profinite ring Λ is plainly a compact, Hausdorff and totally disconnected topological ring; the converse is also true, as we indicate in Proposition 5.1.2 below. It is clear that a profinite ring admits a fundamental system of neighborhoods of 0 consisting of open (two-sided) ideals (this follows from a result analogous to Lemma 2.1.1).

Luis Ribes, Pavel Zalesskii

6. Homology and Cohomology of Profinite Groups

Abstract

In this section we introduce some terminology and sketch some general homological results. We shall state the concepts and results for general abelian categories to avoid repetitions, but we are mainly interested in categories of modules such as Mod(Λ), PMod(Λ) DMod(Λ) or DMod(G), where Λ is a profinite ring and G a profinite group. All functors will be assumed to be additive, i.e., they preserve direct sums systems of the form A ⊕ B.

Luis Ribes, Pavel Zalesskii

7. Cohomological Dimension

Abstract

Let G be a profinite group and let p be a prime number. Recall that if A is an abelian group, then A _p denotes its p-primary component, i.e., the subgroup consisting of those elements of A of order p ⁿ, for some n. If A=A _p we say that A is p-primary. The cohomological p-dimension cd _p(G) of G is the smallest non-negative integer n such that H ^k(G,A)_p=0 for all k>n and $A\in\mathbf {DMod}(\lbrack\!\lbrack\widehat{\mathbf{Z}}G\rbrack\!\rbrack)$, if such an n exists. Otherwise we say that cd _p(G)=∞.

Luis Ribes, Pavel Zalesskii

8. Normal Subgroups of Free Pro - ${\cal C}$ Groups

Abstract

Throughout this chapter $\mathcal{C}$ denotes usually an NE-formation of finite groups, i.e., $\mathcal{C}$ is a nonempty class of finite groups closed under taking normal subgroups, homomorphic images and extensions. Equivalently, $\mathcal{C}$ is the class of all finite Δ-groups, where Δ is a set of finite simple groups (see Section 2.1). In particular, $\mathcal{C}$ could be the class of all finite groups, the class of all finite solvable groups, etc. Often we require in addition that $\mathcal{C}$ ‘involves two different primes’, that is, that there exists a group in $\mathcal{C}$ whose order is divisible by at least two different prime numbers. In this chapter $\varSigma_{\mathcal{C}}$ denotes the collection of all finite simple groups in $\mathcal{C}$, and Σ denotes the class of all finite simple groups.

Luis Ribes, Pavel Zalesskii

9. Free Constructions of Profinite Groups

Abstract

In this chapter we introduce free products, free products with amalgamation and HNN-extensions in the category of pro - ${\cal C}$ groups. We shall study only basic properties of these constructions here.

Luis Ribes, Pavel Zalesskii

Backmatter

Titel: Profinite Groups
verfasst von: Luis Ribes
Pavel Zalesskii
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-01642-4
Print ISBN: 978-3-642-01641-7
DOI: https://doi.org/10.1007/978-3-642-01642-4

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Inverse and Direct Limits

2. Profinite Groups

3. Free Profinite Groups

4. Some Special Profinite Groups

5. Discrete and Profinite Modules

6. Homology and Cohomology of Profinite Groups

7. Cohomological Dimension

8. Normal Subgroups of Free Pro - ${\cal C}$ Groups

9. Free Constructions of Profinite Groups

Backmatter

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