Recíprocas do Teorema de Lagrange

Abstract

Group Theory is widely recognized as the oldest branch of modern algebra. Its origins date back to the pioneering studies of renowned mathematicians, such as Joseph Louis Lagrange (1736-1813), Paolo Ruffini (1765-1822) and Évariste Galois (1811-1832), who stood out in the context of the theory of Algebraic Equations. Among the most notable theorems in Group Theory, Lagrange’s Theorem stands out. Thus, the purpose of this work is to formulate and demonstrate the main reciprocals of this theorem. The converse of Lagrange’s Theorem asks that: if G is a group of order n and d is a divisor of G, then is there a subgroup in G of order d? In general, the answer to this question is no. However, it is essential to highlight that under certain conditions and for specific groups, this reciprocal is true. To achieve the purpose of this work, we chose to carry out research with a qualitative, bibliographical approach, based on books that have a solid structural organization and cohesive language. Initially we provide a review of the introductory concepts about Groups, covering topics such as Subgroups, Group Homomorphism, Cyclic Groups, Permutation Groups, Dihedral Groups, Classes laterals, Nor mal Subgroup, Quotient Group and others. We also detail Lagrange’s Theorem and its demon stration, in order to facilitate the reader’s understanding of the following results. Furthermore, in order to understand and demonstrate the main results regarding the reciprocal of Lagrange’s Theorem, which are the three Sylow Theorems, with emphasis on the first, and Cauchy’s Theo rem, other more advanced tools were presented and used, such as, for example, Group Actions, Conjugacy Classes, Class Equation, Orbit, Stabilizer and P-Groups.