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.