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 homolยญ 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 disยญ 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 propยญ erties of G from corresponding properties of its finite homomorphic images
CONTENT
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-C Groups -- 9 Free Constructions of Profinite Groups -- Open Questions -- A1 Spectral Sequences -- A2 Positive Spectral Sequences -- A3 Spectral Sequence of a Filtered Complex -- A4 Spectral Sequences of a Double Complex -- A5 Notes, Comments and Further Reading -- Index of Symbols -- Index of Authors -- Index of Terms
