An introduction to profinite groups

Jean Lecureux

Description

Niveau: Supérieur, Doctorat, Bac+8
An introduction to profinite groups Jean Lécureux February 20, 2006 These notes are intended to give a general survey about profinite groups, following [2]. While the two first parts explain the definitions and basic properties of profinite groups, the last one is intended to show how profinite groups could appear in a concrete group theoretical problem. Notation. If N is a subgroup of a topological group G, we write N < G. If N is normal, we write N / G. If N is open (resp. closed), we add a o (resp. c), to get something like N /o G. 1 Inverse limits Definition 1.1. A directed set is a partially ordered set (?,≤) such that for every ?, µ ? ?, there exists some ? ? ? such that ? ≥ ? and ? ≥ µ. Example 1.2. A totally ordered set is a directed set. Example 1.3. N? with the order given by the divisibility is a directed set. Definition 1.4. An inverse systems over a directed set ? is a family of sets (or groups, or rings, or topological spaces, or anything) G? with maps (or homomorphism, or continuous maps, or morphisms of anything), defined when ? ≥ µ, pi?µ : G? ? Gµ such that pi?? = IdG? and piµ? ? pi?µ = pi?? .

• problem can


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• can get


• hausdor?


• compact hausdor? topological


• finitely many


• subgroup

Sujets

• Roger Lecureux
• Finite set
• Subgroup

Informations

• Publié par: mijec
• Nombre de lectures: 43
• Langue: English

Extrait

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