The Kurosh Subgroup Theorem for profinite groups

• Autor(a) principal: Mattheus Pereira da Silva Aguiar
• Data de Publicação: 2019
• Tipo de documento: Dissertação
• Idioma: por
• Título da fonte: Repositório Institucional da UFMG
• Texto Completo: http://hdl.handle.net/1843/31756
• topic: Profinite graphs; Profinite groups; Galois coverings; Fundamental group; Nielsen- Schreier; Graph of pro-C groups; Kurosh; Grupos profinitos; Galois, Teoria de; Teoria dos grupos; Grupos fundamentais (Matemática).; Teoria dos grafos

A profinite graph is a profinite space with a graph structure, i.e., a closed set named the vertex set and two continuous incidence maps d0; d1 V (). A profinite group G can act on a profinite graph and G is called the quotient graph by the action of G. If G acts freely, the map G is called a Galois covering of the profinite graph and the group G the associated group of this Galois covering. We can also define a universal Galois covering, where the associated group with this covering is the fundamental group of the profinite graph , denoted by C1 . We give an original proof of the Nielsen-Schreier Theorem for free profinite groups over a finite space, which states that every open profinite subgroup of a free profinite group on a finite space is a free profinite subgroup on a finite space using these aforementioned structures. We also define a sheaf of pro-C groups, the free pro-C product of a sheaf and use it to define a graph of pro-C groups and the fundamental group of a graph of pro-C groups in a similar manner we did previously. The analogue of the universal Galois covering is named the standard graph, denoted by SC(G;). Finally we show that the free product of pro-C groups can be seen as the fundamental group of a graph of pro-C groups and use this fact to prove the Kurosh Subgroup Theorem for profinite groups, the main result of this thesis.