On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups

Let G be a finite group and 𝒩𝒞(G) the set of the numbers of conjugates of noncyclic proper subgroups of G. We prove that (1) if |𝒩𝒞(G)| ≤ 2, then G is solvable, and (2) G is a nonsolvable group with |𝒩𝒞(G)| = 3 if and only if G≅PSL(2,5) or PSL(2,13) or SL(2,5) or SL(2,13).

Detalles Bibliográficos
Autores principales: Shi, Jiangtao, Zhang, Cui
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Hindawi Publishing Corporation 2013
Materias:
Research Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3826299/
https://www.ncbi.nlm.nih.gov/pubmed/24292508
http://dx.doi.org/10.1155/2013/430870
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author Shi, Jiangtao
Zhang, Cui
author_facet Shi, Jiangtao
Zhang, Cui
author_sort Shi, Jiangtao
collection PubMed
description Let G be a finite group and 𝒩𝒞(G) the set of the numbers of conjugates of noncyclic proper subgroups of G. We prove that (1) if |𝒩𝒞(G)| ≤ 2, then G is solvable, and (2) G is a nonsolvable group with |𝒩𝒞(G)| = 3 if and only if G≅PSL(2,5) or PSL(2,13) or SL(2,5) or SL(2,13).
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spelling pubmed-38262992013-11-28 On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups Shi, Jiangtao Zhang, Cui ScientificWorldJournal Research Article Let G be a finite group and 𝒩𝒞(G) the set of the numbers of conjugates of noncyclic proper subgroups of G. We prove that (1) if |𝒩𝒞(G)| ≤ 2, then G is solvable, and (2) G is a nonsolvable group with |𝒩𝒞(G)| = 3 if and only if G≅PSL(2,5) or PSL(2,13) or SL(2,5) or SL(2,13). Hindawi Publishing Corporation 2013-10-27 /pmc/articles/PMC3826299/ /pubmed/24292508 http://dx.doi.org/10.1155/2013/430870 Text en Copyright © 2013 J. Shi and C. Zhang. https://creativecommons.org/licenses/by/3.0/ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
spellingShingle Research Article
Shi, Jiangtao
Zhang, Cui
On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups
title On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups
title_full On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups
title_fullStr On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups
title_full_unstemmed On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups
title_short On the Set of the Numbers of Conjugates of Noncyclic Proper Subgroups of Finite Groups
title_sort on the set of the numbers of conjugates of noncyclic proper subgroups of finite groups
topic Research Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3826299/
https://www.ncbi.nlm.nih.gov/pubmed/24292508
http://dx.doi.org/10.1155/2013/430870
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