Grupos solúveis e grupos nilpotentes

Abstract

In this work we will study some concepts related to soluble groups and groups nilpotent. One of the oldest concepts in the study of group theory is that of soluble groups, which was developed by the famous mathematician Evariste Galois. A group G is said to be soluble if it contains a chain of subgroups{1} = G0 ≤ G1 ≤ . . . ≤ Gn = G, such that each sub-group Gi−1 is normal in Gi and the quotients groups Gi/Gi−1, 1 ≤ i ≤ n are abelian. The conditions required in the definition of nilpotency are more demanding than the definitions of soluble groups. So, a group is said to be nilpotent if it contains a series of subgroups such that each subgroup Gi−1 is normal in G and each quotient Gi/Gi−1 is contained in the center of G/Gi−1, 1 i n. This series of subgroups is called a central series of G, having as examples the upper central series ({1} = ζ0(G) ≤ ζ1(G) ≤ · · · ζn(G) ≤ · · · ) and the lower central series (G = γ1(G) ≥ γ2(G) ≥· · · ≥ γn(G) ≥ · · · ). To reach the objective of the study, we will present some basic definitions such as groups, permutation groups, subgroups, classes laterals, Lagrange theorem, normal subgroups, quotient group and homomorphism. Sylow’s theorems will also be discussed, as they guarantee us certain results for the groups nilpotent.