Browsing by Subject "topologiset ryhmät"
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(2024)In this thesis, we aim to introduce the reader to profinite groups. Profinite groups are defined by two characteristics: firstly, they have a topology defined on them (notably, they are compact). Secondly, they are constructed from some collection of finite groups, each equipped with a discrete topology and forming what is known as an inverse system. The profinite group emerges as an inverse limit of its constituent groups. This definition is, at this point, necessarily quite abstract. Thus, before we can really understand profinite groups we must examine two areas: first, we will study topological groups. This will give us the means to deal with groups as topological spaces. Topological groups have some characteristics that differentiate them from general topological spaces: in particular, a topological group is always a homogeneous space. Secondly, we will explore inverse systems and inverse limits, which will take us into category theory. While we could explain these concepts without categories, this thesis takes the view that category theory gives us a useful “50000feet view” by giving these ideas a wider mathematical context. In the second chapter, we will go through preliminary information concerning group theory, general topology and category theory that will be needed later. We will begin with some basic concepts from group theory and pointset topology. These sections will mostly contain information that is familiar from the introductory university courses. The chapter will then continue by introducing some basic concepts of category theory, including inverse systems and inverse limits. For these, we will give an application by showing how the Cantor set is homeomorphic to an inverse limit of a collection of finite sets. In the third chapter, we will examine topological groups and prove some of their properties. In the fourth chapter, we will introduce an example of profinite groups: Zp, the additive group of padic integers. This will be expanded into a ring and then into the field Qp. We will discuss the uses of Zp and Qp and show how to derive them as an inverse limit of finite, compact groups.
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