An Introduction to p-adic Analytic Groups

Abstract

In this thesis we introduce the concept of a p-adic analytic pro-p group. To do so we discuss important theorems and concepts connected to more general types of groups, these include profinite groups, pro-p groups, powerful p-groups, powerful pro-p groups and uniform groups.

Popular Abstract

There are few concepts that are as integral to our culture, thinking and to the nature of the universe as the idea of symmetry. From the most fundamental principles of physics to the simple reflection symmetry of a human face, symmetry can be found in the smallest of particles and the biggest astronomical objects. When thinking about symmetry the concepts of reflection symmetry or rotational symmetry immediately come to mind and they suggest that objects are symmetric if they do not change their structure when a certain action is carried out. We usually think of this action as a rotation or reflection but in a mathematical sense this action is less restricted. If we consider four identical objects that are placed in a line we can certainly perform a reflection and thus exchange the first and the fourth as well as the second and the third object. However, would not the structure remain the same if we exchanged the first and the second object for example? Thus, we have discovered another symmetry. This is just a very simple example of a symmetry in a mathematical sense, however, symmetries can be extremely complex.

We can now define, in a very intuitive way, what a group is. Let us say that we are given a structure with a certain symmetry. This can be something as simple as the four objects from above. Now we consider all actions that leave the structure the same, in other words all actions related to a certain symmetry, let us call these actions symmetry actions. Then this collection of symmetry actions is a group. Not only that but every group can be seen as such a collection of symmetry actions for some structure (Frucht's theorem).

Let us now have a closer look at the structure of four identical objects. A very important thing to notice is that we can perform two such symmetry actions successively. Let us consider the action of exchanging the first and second object and let us denote this symmetry action as (1 2). So what happens if we first carry out this action and afterwards exchange the second and third object, (2 3)? Well, the first object gets moved to the second position and then to the third position. The second objects is moved to the first position and the third object is moved to the second position. Now, what happens if we exchange the order of the two symmetry actions that we have performed? Firstly, we exchange the second and the third object and secondly, we exchange the first and the second object. Then the third object ends up in the first position, while the first object moves to the second position and the second object moves to the third position. In other words, the outcome is not the same; indeed, (1 2)(2 3) is not equal to (2 3)(1 2)! Thus, for some groups we can not change the order of the symmetry actions. However, for many other groups it does not matter in which order we perform the symmetry actions. These groups are called abelian groups. In a certain sense, abelian groups are easier to understand and to work with. For this reason there is a lot of research about them.

Another property that many groups have is that if we take a symmetry action and apply it several times we end up where we started. If we exchange the first and the second object in the example above and then do the exact same symmetry action again we end up at the starting position. In this case we needed to apply the symmetry action (1 2) only twice to arrive at the starting position, we say that (1 2) has order 2. If we consider the symmetry action of rotating a square 90 degrees, we need to carry out this action four times to end up where we started. In other words, the symmetry action of rotating a square 90 degrees has order 4. Let us now consider a certain prime p. A group where every element, i.e. every symmetry action has order of a power of p is called a p-group. Those groups are very well understood and are very important tools when it comes to understanding groups in general.

In this thesis I will introduce different types of groups. In general we want to study groups that have a useful structure for example to classify certain types of groups. We also want to achieve a high level of understanding of the types of groups that we are looking at. For those two reason it is essential to look at groups with a relatively simple structure and then generalize the findings to more complex groups. I will do this on two separate occasions in this thesis.

Firstly, the easiest method to separate "easy" groups from "hard" groups is by dividing them into finite and infinite groups. Looking at a finite amount of symmetry actions is in most cases an easier job than looking at an infinite amount of such actions. Having that in mind I will first introduce the concept of a "profinite group". These groups are more or less a combination of finite groups and a logical generalization of the concept of finiteness.

In a similar manner I will introduce "powerful" p-groups. These are a generalization of the well-understood abelian p-groups.

Finally this will lead me to groups that are both profinite and powerful p-groups. These are called powerful pro-p groups. The p-adic analytic pro-p groups are a special case of such powerful pro-p groups and of particular interest to group theoretic research since they have been useful when proving or disproving certain conjectures.

• author: Selisko, Frieder
• supervisor: Anitha Thillaisundaram
• organization: Mathematics (Faculty of Engineering); Mathematics (Faculty of Sciences); Centre for Mathematical Sciences
• course: MATK11 20231
• year: 2023
• type: M2 - Bachelor Degree
• subject: Mathematics and Statistics
• publication/series: Bachelor's Theses in Mathematical Sciences
• report number: LUNFMA-4144-2023
• ISSN: 1654-6229
• other publication id: 2023:K14
• language: English
• id: 9121788
• date added to LUP: 2023-06-19 14:06:12
• date last changed: 2023-06-19 14:06:12