Finite groups with all minimal subgroups solitary

We give a complete classification of the finite groups with a unique subgroup of order p for each prime p dividing its order. All the groups considered in this paper will be finite. One of the most fruitful lines in the research in abstract group theory during the last years has been the study of gr...

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Authors: Esteban Romero, Ramón, Liriano, Orieta
Format: article
Publication Date:2016
Country:España
Institution:Universitat Politècnica de València (UPV)
Repository:RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia
Language:English
OAI Identifier:oai:riunet.upv.es:10251/84110
Online Access:https://riunet.upv.es/handle/10251/84110
Access Level:Open access
Keyword:Finite group
Solitary subgroup
Minimal subgroup
MATEMATICA APLICADA
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spelling Finite groups with all minimal subgroups solitaryEsteban Romero, RamónLiriano, OrietaFinite groupSolitary subgroupMinimal subgroupMATEMATICA APLICADAWe give a complete classification of the finite groups with a unique subgroup of order p for each prime p dividing its order. All the groups considered in this paper will be finite. One of the most fruitful lines in the research in abstract group theory during the last years has been the study of groups in which the members of a certain family of subgroups satisfy a certain subgroup embedding property. The family of the subgroups of prime order (also called minimal subgroups) has attracted the interest of many mathematicians. For example, a well-known result of Itˆo (see [8, Kapitel III, Satz 5.3; 9]) states that a group of odd order with all minimal subgroups in the center is nilpotent. The following result of Gasch¨utz and Itˆo (see [5, Kapitel IV, Satz 5.7; 9]) gives interesting properties of groups with all minimal subgroups normal.The first author has been supported by the research grant MTM2014-54707-C03-1-P from the Ministerio de Economia y Competitividad, Spain and FEDER, European Union. The research of the second author has been done during some visits to the Departament de Matematica Aplicada and the Institut de Matematica Pura i Aplicada of the Universitat Politecnica de Valencia and the Departament d' Algebra of the Universitat de Valencia. She expresses her gratitude to these institutions for the use of their facilities and, especially to the first university for the financial support of some of the visits.World Scientific PublishingMinisterio de Economía y CompetitividadUniversitat de ValènciaUniversitat Politècnica de ValènciaRepositorio Institucional de la Universitat Politècnica de València Riunet20162016-10-01journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfapplication/pdfhttps://riunet.upv.es/handle/10251/84110reponame:RiuNet. Repositorio Institucional de la Universitat Politécnica de Valénciainstname:Universitat Politècnica de València (UPV)InglésengMinisterio de Economía y Competitividad http://dx.doi.org/10.13039/501100003329 MTM2014-54707-C3-1-P PROPIEDADES ARITMETICAS Y ESTRUCTURALES DE GRUPOS Y SEMIGRUPOS Iopen accesshttp://purl.org/coar/access_right/c_abf2Reserva de todos los derechoshttp://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:riunet.upv.es:10251/841102026-06-13T07:49:27Z
dc.title.none.fl_str_mv Finite groups with all minimal subgroups solitary
title Finite groups with all minimal subgroups solitary
spellingShingle Finite groups with all minimal subgroups solitary
Esteban Romero, Ramón
Finite group
Solitary subgroup
Minimal subgroup
MATEMATICA APLICADA
title_short Finite groups with all minimal subgroups solitary
title_full Finite groups with all minimal subgroups solitary
title_fullStr Finite groups with all minimal subgroups solitary
title_full_unstemmed Finite groups with all minimal subgroups solitary
title_sort Finite groups with all minimal subgroups solitary
dc.creator.none.fl_str_mv Esteban Romero, Ramón
Liriano, Orieta
author Esteban Romero, Ramón
author_facet Esteban Romero, Ramón
Liriano, Orieta
author_role author
author2 Liriano, Orieta
author2_role author
dc.contributor.none.fl_str_mv Ministerio de Economía y Competitividad
Universitat de València
Universitat Politècnica de València
Repositorio Institucional de la Universitat Politècnica de València Riunet
dc.subject.none.fl_str_mv Finite group
Solitary subgroup
Minimal subgroup
MATEMATICA APLICADA
topic Finite group
Solitary subgroup
Minimal subgroup
MATEMATICA APLICADA
description We give a complete classification of the finite groups with a unique subgroup of order p for each prime p dividing its order. All the groups considered in this paper will be finite. One of the most fruitful lines in the research in abstract group theory during the last years has been the study of groups in which the members of a certain family of subgroups satisfy a certain subgroup embedding property. The family of the subgroups of prime order (also called minimal subgroups) has attracted the interest of many mathematicians. For example, a well-known result of Itˆo (see [8, Kapitel III, Satz 5.3; 9]) states that a group of odd order with all minimal subgroups in the center is nilpotent. The following result of Gasch¨utz and Itˆo (see [5, Kapitel IV, Satz 5.7; 9]) gives interesting properties of groups with all minimal subgroups normal.
publishDate 2016
dc.date.none.fl_str_mv 2016
2016-10-01
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://riunet.upv.es/handle/10251/84110
url https://riunet.upv.es/handle/10251/84110
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Ministerio de Economía y Competitividad http://dx.doi.org/10.13039/501100003329 MTM2014-54707-C3-1-P PROPIEDADES ARITMETICAS Y ESTRUCTURALES DE GRUPOS Y SEMIGRUPOS I
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Reserva de todos los derechos
http://rightsstatements.org/vocab/InC/1.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Reserva de todos los derechos
http://rightsstatements.org/vocab/InC/1.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv World Scientific Publishing
publisher.none.fl_str_mv World Scientific Publishing
dc.source.none.fl_str_mv reponame:RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia
instname:Universitat Politècnica de València (UPV)
instname_str Universitat Politècnica de València (UPV)
reponame_str RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia
collection RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia
repository.name.fl_str_mv
repository.mail.fl_str_mv
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