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Previous issue date: 2013-07-05 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This document presents an approach and development of some of the results of
Shumyatsky in [14, 15, 16, 17, 18], where he worked with automorphisms of order two
in finite groups of odd order, mainly showing the influence that the structure of the
centralizer has on that of Group. Let G be a group with odd order, and ϕ an automorphism
on G, of order two, where G = [G,ϕ], and given a limitation in the order of the centralizer
of ϕ regard to G, CG(ϕ), which induces a limitation in the order of derived group G′ of
group G, and we also verified that G has a normal subgroup H that is ϕ-invariant, such
that H′ ≤ Gϕ and its index [G : H] is bounded with the initial limitation. With the same
hypothesis of the group G and with the same limitation of the order of the centralizer of
the automorphism, let V a abelian p-group such that G⟨ϕ⟩ act faithful and irreductible
on V, then there is a bounded constant k, limitated by a function depending only on the
parameter m, where m is tha limitation in the order of CG(ϕ), and elements x1, ...xk ∈ G−ϕ
such that V = ρϕx
1,...,xk(V−ϕ). / O trabalho baseia-se na apresentação e desenvolvimento de alguns resultados expostos
por Shumyatsky em [14, 15, 16, 17, 18], onde trabalha com automorfismos de ordem
dois em grupos de ordem ímpar, mostrando fundamentalmente a influência da estrutura
do centralizador do automorfismo na estrutura do grupo. Seja G um grupo de ordem
ímpar e ϕ um automorfismo de G, de ordem dois, tal que G = [G,ϕ], dada uma limitação
na ordem do centralizador de ϕ em G, CG(ϕ), a mesma induz uma limitação na ordem do
grupo derivado G′ do grupo G, além disso verificamos que G tem um subgrupo H normal
ϕ-invariante, tal que H′ ≤ Gϕ e o índice [G : H] é limitado dependendo da limitação
inicial de CG(ϕ). Nas mesmas hipóteses do grupo G e com a mesma limitação da ordem
do centralizador do automorfismo, seja V um p-grupo abeliano, tal que G⟨ϕ⟩ age fiel e
irredutivelmente sobre V, então existe uma constante k, limitada por uma função que
depende só da limitação de CG(ϕ), e elementos x1, ...xk ∈ G−ϕ, tal que V = ρϕx
1,...,xk(V−ϕ).
| Identifer | oai:union.ndltd.org:IBICT/oai:repositorio.bc.ufg.br:tede/3090 |
| Date | 05 July 2013 |
| Creators | Rojas, Yerko Contreras |
| Contributors | Lima, Aline de Souza |
| Publisher | Universidade Federal de Goiás, Programa de Pós-graduação em Matemática (IME), UFG, Brasil, Instituto de Matemática e Estatística - IME (RG) |
| Source Sets | IBICT Brazilian ETDs |
| Language | Portuguese |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, info:eu-repo/semantics/masterThesis |
| Format | application/pdf |
| Source | reponame:Biblioteca Digital de Teses e Dissertações da UFG, instname:Universidade Federal de Goiás, instacron:UFG |
| Rights | http://creativecommons.org/licenses/by-nc-nd/4.0/, info:eu-repo/semantics/openAccess |
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