Profinite groups are of interest to mathematicians working in a variety of areas, including number theory, abstract groups, and analysis. The underlying theory reflects these diverse influences, with methods drawn from both algebra and topology and with fascinating connections to field theory. This is the first book to be dedicated solely to the study of general profinite groups. It provides a thorough introduction to the subject, designed not only to convey the basic facts but also to enable readers to enhance their skills in manipulating profinite groups. The first few chapters lay the foundations and explain the role of profinite groups in number theory. Later chapters explore various aspects of profinite groups in more detail; these contain accessible and lucid accounts of many major theorems. Prerequisites are kept to a minimum with the basic topological theory summarized in an introductory chapter.

• 0: Topological preliminaries


• 1: Profinite groups and completions


• 2: Sylow theory


• 3: Galois theory


• 4: Finitely generated groups and countably based groups


• 5: Free groups and projective groups


• 6: Modules, extensions, and duality


• 7: Modules for completed group algebras


• 8: Profinite groups of finite rank


• 9: Cohomology of profinite groups


• 10: Further cohomological methods


• 11: Groups of finite cohomological dimension


• 12: Finitely presented pro-p groups