Profinite Groups

In this thesis we will give a self-contained exposition of profinite groups, from definition to key properties, focusing on their topology and generators, the generalisation of the Sylow Theorems and how they may be seen as group completions. Each section will present motivating examples most relevant to modern research in number theory, such as the p-adic integers, the profinite integers, the Nottingham group and infinite Galois groups. Since the construction of profinite groups relies on inverse limits, we will analyse them in detail, introducing the necessary category theoretical notions. We will explore at length the link between infinite Galois theory and profinite groups, generalising the Fundamental Theorem and proving the realisability of certain profinite groups as Galois groups.

Thesis