Profinite groups

Abstract

This thesis provides a comprehensive study of profinite groups, which
are fascinating mathematical objects that have attracted significant interest in modern algebraic research. Profinite groups are infinite generalizations of finite groups and share many similarities with them. They are endowed with a topology that makes them compact and totally disconnected,
which is the foundation from which we draw conclusions on their structure.
In this thesis, we introduce the basic concepts of profinite groups and their
relationship with finite groups. We then generalize Lagrange’s theorem
and the Sylow theorems to profinite groups, which are crucial for understanding their structure and subgroups. We also provide a generalization
of the Fundamental Theorem of Galois Theory to profinite groups, which
has important implications for the study of number theory and algebraic
geometry. We conclude with examples of infinite Galois groups, including
the Galois group of the algebraic closure of finite fields, and some infinite
Galois extensions of the rational numbers, which illustrate the power of
profinite groups in studying the structure of infinite Galois groups.

Popular Abstract

This thesis provides a comprehensive study of a special kind of
topological groups. Topological groups are sets with a notion of
“closeness” defined by their topological structure, and with a binary
operation between the elements in the set (an operation that takes
two elements in the set to a new element in the set) that satisfies
a compatibility condition that makes the binary operation map
“close” elements to “close” elements. These concepts serve as a
bridge between fields of study in mathematics such as topology and
geometry, and are used to describe and analyze physical systems in
quantum mechanics and particle physics, being key to understanding
continuous symmetries.
Profinite groups are a special type of possibly infinite abstract
topological groups. These groups are constructed from collections of
finite groups in such a way that many of the properties related to
the finite groups in a collection are inherited by their profinite group.
This construction is called an inverse limit, which is a mathematical
structure that allows us to “glue” together the finite groups to form
a possibly infinite group.
Due to this relationship between finite groups and profinite groups
many theorems relating finite groups can be extended to profinite
groups. One such theorem —and the motivation from which profinite
groups arose— is the Fundamental Theorem of Galois Theory, an
extremely powerful result that establishes a deep connection between
group theory and field theory.
In this thesis we explore all of these concepts and topics in
hopes of gaining a better understanding of these mathematical structures. We study profinite groups as inverse limits of finite groups; as
topological groups with specific properties, namely compactness and
total disconnectedness; and as Galois groups, the object of study of
Galois Theory.

• author: Moreno Poncela, Juan
• supervisor: Anitha Thillaisundaram
• organization: Mathematics (Faculty of Sciences); Centre for Mathematical Sciences
• course: MATM03 20231
• year: 2023
• type: H2 - Master's Degree (Two Years)
• subject: Mathematics and Statistics
• keywords: Profinite Groups, Infinite Galois Theory
• publication/series: Master's Theses in Mathematical Sciences
• report number: LUNFMA-3138-2023
• ISSN: 1404-6342
• other publication id: 2023:E63
• language: English
• id: 9130701
• date added to LUP: 2023-07-11 15:08:31
• date last changed: 2023-07-11 15:08:31