Profinite Groups

1. Front Matter

   Pages I-XIV

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2. Inverse and Direct Limits

   • Luis Ribes, Pavel Zalesskii

   Pages 1-18

3. Profinite Groups

   • Luis Ribes, Pavel Zalesskii

   Pages 19-77

4. Free Profinite Groups

   • Luis Ribes, Pavel Zalesskii

   Pages 79-121

5. Some Special Profinite Groups

   • Luis Ribes, Pavel Zalesskii

   Pages 123-163

6. Discrete and Profinite Modules

   • Luis Ribes, Pavel Zalesskii

   Pages 165-200

7. Homology and Cohomology of Profinite Groups

   • Luis Ribes, Pavel Zalesskii

   Pages 201-257

8. Cohomological Dimension

   • Luis Ribes, Pavel Zalesskii

   Pages 259-300

9. Normal Subgroups of Free Pro-C Groups

   • Luis Ribes, Pavel Zalesskii

   Pages 301-360

10. Free Constructions of Profinite Groups

    • Luis Ribes, Pavel Zalesskii

    Pages 361-400

11. Back Matter

    Pages 401-435

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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. In neither of these two aspects have we tried to be encyclopedic. After some necessary background, we thoroughly develop the basic properties of profinite groups and introduce the main tools of the subject in algebra, topology and homol­ ogy. Later we concentrate on some topics that we present in detail, including recent developments in those areas. Interest in profinite groups arose first in the study of the Galois groups of infinite Galois extensions of fields. Indeed, profinite groups are precisely Galois groups and many of the applications of profinite groups are related to number theory. Galois groups carry with them a natural topology, the Krull topology. Under this topology they are Hausdorff compact and totally dis­ connected topological groups; these properties characterize profinite groups. Another important fact about profinite groups is that they are determined by their finite images under continuous homomorphisms: a profinite group is the inverse limit of its finite images. This explains the connection with abstract groups. If G is an infinite abstract group, one is interested in deducing prop­ erties of G from corresponding properties of its finite homomorphic images.

• Galois group
• Profinite group
• cohomology
• number theory
• topological group
• topology

• School of Mathematics and Statistics, Carleton University, Ottawa, Canada

  Luis Ribes

• Department of Mathematics, University of Brasilia, Brasilia, Brazil

  Pavel Zalesskii

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