Sobre a influência dos centralizadores dos automorfismos de ordem dois em grupos de ordem ímpar

Detalhes bibliográficos
Autor(a) principal: Rojas, Yerko Contreras
Data de Publicação: 2013
Tipo de documento: Dissertação
Idioma: por
Título da fonte: Repositório Institucional da UFG
dARK ID: ark:/38995/001300000f2sn
Texto Completo: http://repositorio.bc.ufg.br/tede/handle/tede/3090
Resumo: 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).
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spelling Lima, Aline de Souzahttp://lattes.cnpq.br/1518865173435209http://lattes.cnpq.br/9263250131156705Rojas, Yerko Contreras2014-09-18T15:43:59Z2013-07-05ROJAS, Yerko Contreras. Sobre a Influência dos Centralizadores dos Automorfismos de Ordem Dois em Grupos de Ordem Ímpar. 2013. 59 f. Dissertação (Mestrado em Matemática ) - Universidade Federal de Goiás, Goiânia, 2013.http://repositorio.bc.ufg.br/tede/handle/tede/3090ark:/38995/001300000f2snThis 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).Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2014-09-18T15:33:16Z No. of bitstreams: 2 Dissertacao Yerko Contreras Rojas.pdf: 673331 bytes, checksum: 5359343f8c3a32e21369c3bc57917634 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5)Approved for entry into archive by Luciana Ferreira (lucgeral@gmail.com) on 2014-09-18T15:43:59Z (GMT) No. of bitstreams: 2 Dissertacao Yerko Contreras Rojas.pdf: 673331 bytes, checksum: 5359343f8c3a32e21369c3bc57917634 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5)Made available in DSpace on 2014-09-18T15:43:59Z (GMT). No. of bitstreams: 2 Dissertacao Yerko Contreras Rojas.pdf: 673331 bytes, checksum: 5359343f8c3a32e21369c3bc57917634 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2013-07-05Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPESapplication/pdfhttp://repositorio.bc.ufg.br/tede/retrieve/7930/Dissertacao%20Yerko%20Contreras%20Rojas.pdf.jpgporUniversidade Federal de GoiásPrograma de Pós-graduação em Matemática (IME)UFGBrasilInstituto de Matemática e Estatística - IME (RG)[1] ASAR, A. Involutory automorphisms of groups of odd order. Arch. Math.36, p. 97–103, 1981. [2] BURNSIDE, W. Theory of Groups. Dover, New York, 1955. [3] FEIT, W. THOMPSON, J. Solvability of groups of odd order. Pacific J Math 13, p. 773–1029, 1963. [4] GORENSTEIN, D. Finite Groups. Harper and Row, New York, 1968. [5] HARTLEY, B. Periodic locally soluble groups containing an element of prime order with cêrnikov centralizer. Q. J Math, Oxford,2 33, p. 309–323, 1982. [6] HARTLEY, B. MEIXNER, T. Periodic groups in which the centralizer of an involution has bounded order. J Algebra, 64:285–291, 1980. [7] HARTLEY, B. MEIXNER, T. Finite soluble groups in which the centralizer of an element the prime order is small. Arch. Math., 36:211–213, 1981. [8] HIGMAN, G. Groups and rings having automorphisms without non-trivial fixed elements. J. London Math. Soc., 2(32):321–334, 1957. [9] HUNDERFORD, T. Algebra. Springer-Verlag, New York, 1980. [10] ISAACS, M. Finite Group Theory. Volumen 92 de Graduate Studies in Mathematics, American Mathematical Soc, 2008. [11] KHUKHRO, E. Nilpotent Groups and their Automorphisms. de Gruyter-Verlag, Berlin, 1993. [12] KOVACS, L.G. WALL, G. Involutory automorphisms of groups of odd order and their fixed point groups. Nagoya Math J, (27):113–120, 1966. [13] ROTMAN, J. An Introduction to the Theory of Groups. Springer-Verlag, fourth edition, 1994. [14] SHUMYATSKY, P. Involutory automorphisms of locally soluble periodic groups. J Algebra, (155):36–44, 1993. [15] SHUMYATSKY, P. Involutory automorphisms of periodic groups. Internat J Algebra Comput, (6):745–749, 1996. [16] SHUMYATSKY, P. Involutory automorphisms of finite groups and their centralizers. Arch Math (Basel), (71):425–432, 1998. [17] SHUMYATSKY, P. Involutory automorphisms of groups of odd order. Monatsh. Math, (146):77–82, 2005. [18] SHUMYATSKY, P. Centralizer of involutory automorphisms of groups of odd order. Journal of Álgebra, (315):954–962, 2007. [19] THOMPSON, J. Finite groups with fixed point-free-automorphisms of prime order. Proc. Natl. Acad. Sci. U.S.A., (45):578–581, 1959. [20] THOMPSON, J. automorphisms of solvable groups. Journal. Algebra, (1):259–267, 1964. [21] WARD, J. Involutory automorphisms of groups of odd order. J Austral Math Soc, (6):480–494, 1966.6600717948137941247600600600600-4268777512335152015-85775632160526569622075167498588264571http://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessGrupos finitosGrupos nilpotentesCentralizadores de automorfimosAutomorfismos involutivosFinite groupsNilpotent groupsCentralizer of automorphismsInvolutory automorphismsALGEBRA::LOGICA MATEMATICASobre a influência dos centralizadores dos automorfismos de ordem dois em grupos de ordem ímparCentralizers of involutory automorphisms of groups of odd orderinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisreponame:Repositório Institucional da UFGinstname:Universidade Federal de Goiás (UFG)instacron:UFGLICENSElicense.txtlicense.txttext/plain; charset=utf-82165http://repositorio.bc.ufg.br/tede/bitstreams/e847fae2-946f-4807-b995-1d12090f80aa/downloadbd3efa91386c1718a7f26a329fdcb468MD51CC-LICENSElicense_urllicense_urltext/plain; charset=utf-849http://repositorio.bc.ufg.br/tede/bitstreams/cbc4cee1-e080-420a-ae18-2e0117a5388e/download4afdbb8c545fd630ea7db775da747b2fMD52license_textlicense_texttext/html; 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dc.title.por.fl_str_mv Sobre a influência dos centralizadores dos automorfismos de ordem dois em grupos de ordem ímpar
dc.title.alternative.eng.fl_str_mv Centralizers of involutory automorphisms of groups of odd order
title Sobre a influência dos centralizadores dos automorfismos de ordem dois em grupos de ordem ímpar
spellingShingle Sobre a influência dos centralizadores dos automorfismos de ordem dois em grupos de ordem ímpar
Rojas, Yerko Contreras
Grupos finitos
Grupos nilpotentes
Centralizadores de automorfimos
Automorfismos involutivos
Finite groups
Nilpotent groups
Centralizer of automorphisms
Involutory automorphisms
ALGEBRA::LOGICA MATEMATICA
title_short Sobre a influência dos centralizadores dos automorfismos de ordem dois em grupos de ordem ímpar
title_full Sobre a influência dos centralizadores dos automorfismos de ordem dois em grupos de ordem ímpar
title_fullStr Sobre a influência dos centralizadores dos automorfismos de ordem dois em grupos de ordem ímpar
title_full_unstemmed Sobre a influência dos centralizadores dos automorfismos de ordem dois em grupos de ordem ímpar
title_sort Sobre a influência dos centralizadores dos automorfismos de ordem dois em grupos de ordem ímpar
author Rojas, Yerko Contreras
author_facet Rojas, Yerko Contreras
author_role author
dc.contributor.advisor1.fl_str_mv Lima, Aline de Souza
dc.contributor.advisor1Lattes.fl_str_mv http://lattes.cnpq.br/1518865173435209
dc.contributor.authorLattes.fl_str_mv http://lattes.cnpq.br/9263250131156705
dc.contributor.author.fl_str_mv Rojas, Yerko Contreras
contributor_str_mv Lima, Aline de Souza
dc.subject.por.fl_str_mv Grupos finitos
Grupos nilpotentes
Centralizadores de automorfimos
Automorfismos involutivos
topic Grupos finitos
Grupos nilpotentes
Centralizadores de automorfimos
Automorfismos involutivos
Finite groups
Nilpotent groups
Centralizer of automorphisms
Involutory automorphisms
ALGEBRA::LOGICA MATEMATICA
dc.subject.eng.fl_str_mv Finite groups
Nilpotent groups
Centralizer of automorphisms
Involutory automorphisms
dc.subject.cnpq.fl_str_mv ALGEBRA::LOGICA MATEMATICA
description 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).
publishDate 2013
dc.date.issued.fl_str_mv 2013-07-05
dc.date.accessioned.fl_str_mv 2014-09-18T15:43:59Z
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/masterThesis
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status_str publishedVersion
dc.identifier.citation.fl_str_mv ROJAS, Yerko Contreras. Sobre a Influência dos Centralizadores dos Automorfismos de Ordem Dois em Grupos de Ordem Ímpar. 2013. 59 f. Dissertação (Mestrado em Matemática ) - Universidade Federal de Goiás, Goiânia, 2013.
dc.identifier.uri.fl_str_mv http://repositorio.bc.ufg.br/tede/handle/tede/3090
dc.identifier.dark.fl_str_mv ark:/38995/001300000f2sn
identifier_str_mv ROJAS, Yerko Contreras. Sobre a Influência dos Centralizadores dos Automorfismos de Ordem Dois em Grupos de Ordem Ímpar. 2013. 59 f. Dissertação (Mestrado em Matemática ) - Universidade Federal de Goiás, Goiânia, 2013.
ark:/38995/001300000f2sn
url http://repositorio.bc.ufg.br/tede/handle/tede/3090
dc.language.iso.fl_str_mv por
language por
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dc.relation.department.fl_str_mv -4268777512335152015
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dc.relation.sponsorship.fl_str_mv 2075167498588264571
dc.relation.references.por.fl_str_mv [1] ASAR, A. Involutory automorphisms of groups of odd order. Arch. Math.36, p. 97–103, 1981. [2] BURNSIDE, W. Theory of Groups. Dover, New York, 1955. [3] FEIT, W. THOMPSON, J. Solvability of groups of odd order. Pacific J Math 13, p. 773–1029, 1963. [4] GORENSTEIN, D. Finite Groups. Harper and Row, New York, 1968. [5] HARTLEY, B. Periodic locally soluble groups containing an element of prime order with cêrnikov centralizer. Q. J Math, Oxford,2 33, p. 309–323, 1982. [6] HARTLEY, B. MEIXNER, T. Periodic groups in which the centralizer of an involution has bounded order. J Algebra, 64:285–291, 1980. [7] HARTLEY, B. MEIXNER, T. Finite soluble groups in which the centralizer of an element the prime order is small. Arch. Math., 36:211–213, 1981. [8] HIGMAN, G. Groups and rings having automorphisms without non-trivial fixed elements. J. London Math. Soc., 2(32):321–334, 1957. [9] HUNDERFORD, T. Algebra. Springer-Verlag, New York, 1980. [10] ISAACS, M. Finite Group Theory. Volumen 92 de Graduate Studies in Mathematics, American Mathematical Soc, 2008. [11] KHUKHRO, E. Nilpotent Groups and their Automorphisms. de Gruyter-Verlag, Berlin, 1993. [12] KOVACS, L.G. WALL, G. Involutory automorphisms of groups of odd order and their fixed point groups. Nagoya Math J, (27):113–120, 1966. [13] ROTMAN, J. An Introduction to the Theory of Groups. Springer-Verlag, fourth edition, 1994. [14] SHUMYATSKY, P. Involutory automorphisms of locally soluble periodic groups. J Algebra, (155):36–44, 1993. [15] SHUMYATSKY, P. Involutory automorphisms of periodic groups. Internat J Algebra Comput, (6):745–749, 1996. [16] SHUMYATSKY, P. Involutory automorphisms of finite groups and their centralizers. Arch Math (Basel), (71):425–432, 1998. [17] SHUMYATSKY, P. Involutory automorphisms of groups of odd order. Monatsh. Math, (146):77–82, 2005. [18] SHUMYATSKY, P. Centralizer of involutory automorphisms of groups of odd order. Journal of Álgebra, (315):954–962, 2007. [19] THOMPSON, J. Finite groups with fixed point-free-automorphisms of prime order. Proc. Natl. Acad. Sci. U.S.A., (45):578–581, 1959. [20] THOMPSON, J. automorphisms of solvable groups. Journal. Algebra, (1):259–267, 1964. [21] WARD, J. Involutory automorphisms of groups of odd order. J Austral Math Soc, (6):480–494, 1966.
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