This will be an overview of profinite groups. The emphasis will be on the main concepts and properties, and connections with Galois Theory. Some of the properties will be proved, and others simply sketched with clear references for complete proofs.
Here are the topics that will be covered in the three lectures of the course.
1. Motivating example. The Krull topology on a Galois group. The Galois correspondence.
2. Definition(s) of profinite groups. Main properties. Existence of continuous sections.
3. Profinite groups as Galois groups.
4. Order of a profinite group. The Sylow Theorems.
5. Profinite topologies on abstract groups and profinite completions.
6. Free profinite groups. Iwasawa-Mel'nikov characterizations.
7. Free product of profinite groups.
8. Subgroup theorems.
9. Profinite trees.
10. G-modules. Complete group algebra. Complete tensor product.
11. Cohomology and homology of profinite groups.
12. Cohomological dimension.
13. Galois cohomology.
Last modification: 17 January 2012.