Title of the course: Profinite groups
Instructor: Prof. Martin Hils
Institution: University of Münster
Dates: 10-16 August 2026
Prerequisites: Basic knowledge from algebra (in particular elementary group theory, ideally including Sylow Theorems) and some basic point set topology. For the last 1-2 lectures, basic knowledge from Galois theory is needed. (Although, understanding the statement of the main theorem of Galois theory, without proof, will be enough.)
Level: Advanced undergraduate
Abstract: Profinite groups are by definition projective limits of finite groups. They are topological groups, where the topology, called Krull topology, is inherited from the discrete topology on the finite groups they are built from. Historically, they arised in particular as Galois groups of infinite extensions, but they appear in other areas of mathematics as well. In the course, we will give a basic introduction to the theory of profinite groups, and we plan to cover the following:
* basic properties of profinite groups
* topological characterization of profinite group
* Sylow theory
* profinite completions of groups
* procyclic groups
* Pontryagin duality – the special case of (abelian) profinite groups and (abelian) discrete torsion groups
* Galois correspondence for infinite extensions (using as a black box the main theorem of Galois theory for finite extensions)
Language: EN
Textbook: We will mainly use the first sections of the book “Profinite groups” by Luis Ribes and Pavel Zalesskii (Second edition, Springer 2010).