Os teoremas de Sylow

Abstract

This work presents central results in Group Theory and develops a study of Sylow’s Theorems in the context of Abstract Algebra. Initially, the fundamental concepts related to the structure of groups are introduced, including definitions, basic properties, subgroups, and cosets. Based on these concepts, Lagrange’s Theorem and some of its consequences for the study of finite groups are discussed. Normal subgroups, quotient groups, and homomorphisms are also addressed. Next, the concept of group actions on sets is introduced, an essential tool for proving and understanding the three Sylow Theorems, allowing a deeper analysis of the internal structure of groups. Based on this theoretical framework, the work investigates the conditions for the existence, conjugacy, and number of Sylow p-subgroups. Thus, Sylow’s Theorems prove to be fundamental to the development of Group Theory, significantly extending the results obtained from Lagrange’s Theorem and contributing to the study of abstract algebraic structures.