Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes
A group G is said to be a (PF)C-group or to have polycyclic-by-finite conjugacy classes, if G/C_{G}(x^{G}) is a polycyclic-by-finite group for all xin G. This is a generalization of the familiar property of being an FC-group. De Falco et al. (respectively, de Giovanni and Trombetti) studied groups w...
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| Format: | Article |
| Language: | English |
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University of Isfahan
2016-09-01
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| Series: | International Journal of Group Theory |
| Subjects: | |
| Online Access: | http://www.theoryofgroups.ir/article_8776_ca0b92d4179fde3b3ca79f8b4a3ed6ce.pdf |
| _version_ | 1828292181599715328 |
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| author | Mounia Bouchelaghem Nadir Trabelsi |
| author_facet | Mounia Bouchelaghem Nadir Trabelsi |
| author_sort | Mounia Bouchelaghem |
| collection | DOAJ |
| description | A group G is said to be a (PF)C-group or to have polycyclic-by-finite conjugacy classes, if G/C_{G}(x^{G}) is a polycyclic-by-finite group for all xin G. This is a generalization of the familiar property of being an FC-group. De Falco et al. (respectively, de Giovanni and Trombetti) studied groups whose proper subgroups of infinite rank have finite (respectively, polycyclic) conjugacy classes. Here we consider groups whose proper subgroups of infinite rank are (PF)C-groups and we prove that if G is a group of infinite rank having a non-trivial finite or abelian factor group and if all proper subgroups of G of infinite rank are (PF)C-groups, then so is G. We prove also that if G is a locally soluble-by-finite group of infinite rank which has no simple homomorphic images of infinite rank and whose proper subgroups of infinite rank are (PF)C-groups, then so are all proper subgroups of G. |
| first_indexed | 2024-04-13T11:04:28Z |
| format | Article |
| id | doaj.art-72de23a17efa4bbda0687d6f299f1819 |
| institution | Directory Open Access Journal |
| issn | 2251-7650 2251-7669 |
| language | English |
| last_indexed | 2024-04-13T11:04:28Z |
| publishDate | 2016-09-01 |
| publisher | University of Isfahan |
| record_format | Article |
| series | International Journal of Group Theory |
| spelling | doaj.art-72de23a17efa4bbda0687d6f299f18192022-12-22T02:49:19ZengUniversity of IsfahanInternational Journal of Group Theory2251-76502251-76692016-09-015361678776Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classesMounia Bouchelaghem0Nadir Trabelsi1University Setif 1University Setif 1A group G is said to be a (PF)C-group or to have polycyclic-by-finite conjugacy classes, if G/C_{G}(x^{G}) is a polycyclic-by-finite group for all xin G. This is a generalization of the familiar property of being an FC-group. De Falco et al. (respectively, de Giovanni and Trombetti) studied groups whose proper subgroups of infinite rank have finite (respectively, polycyclic) conjugacy classes. Here we consider groups whose proper subgroups of infinite rank are (PF)C-groups and we prove that if G is a group of infinite rank having a non-trivial finite or abelian factor group and if all proper subgroups of G of infinite rank are (PF)C-groups, then so is G. We prove also that if G is a locally soluble-by-finite group of infinite rank which has no simple homomorphic images of infinite rank and whose proper subgroups of infinite rank are (PF)C-groups, then so are all proper subgroups of G.http://www.theoryofgroups.ir/article_8776_ca0b92d4179fde3b3ca79f8b4a3ed6ce.pdfPolycyclic-by-finite conjugacy classesminimal non-(PF)C-groupminimal non-FC-groupPrüfer rank |
| spellingShingle | Mounia Bouchelaghem Nadir Trabelsi Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes International Journal of Group Theory Polycyclic-by-finite conjugacy classes minimal non-(PF)C-group minimal non-FC-group Prüfer rank |
| title | Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes |
| title_full | Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes |
| title_fullStr | Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes |
| title_full_unstemmed | Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes |
| title_short | Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes |
| title_sort | groups whose proper subgroups of infinite rank have polycyclic by finite conjugacy classes |
| topic | Polycyclic-by-finite conjugacy classes minimal non-(PF)C-group minimal non-FC-group Prüfer rank |
| url | http://www.theoryofgroups.ir/article_8776_ca0b92d4179fde3b3ca79f8b4a3ed6ce.pdf |
| work_keys_str_mv | AT mouniabouchelaghem groupswhosepropersubgroupsofinfiniterankhavepolycyclicbyfiniteconjugacyclasses AT nadirtrabelsi groupswhosepropersubgroupsofinfiniterankhavepolycyclicbyfiniteconjugacyclasses |