A note on Hall S-permutably embedded subgroups of finite groups
Let \(G\) be a finite group. Recall that a subgroup \(A\) of \(G\) is said to permute with a subgroup \(B\) if \(AB=BA\). A subgroup \(A\) of \(G\) is said to be \(S\)-quasinormal or \(S\)-permutable in \(G\) if \(A\) permutes with all Sylow subgroups of \(G\). Recall also that \(H^{s G}\) is th...
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| Дата: | 2017 |
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| Формат: | Стаття |
| Мова: | English |
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Lugansk National Taras Shevchenko University
2017
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| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/148 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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oai:ojs.admjournal.luguniv.edu.ua:article-1482017-07-02T21:58:40Z A note on Hall S-permutably embedded subgroups of finite groups Sinitsa, Darya \(S\)-permutable subgroup, Hall \(S\)-permutably embedded subgroup, \(S\)-permutable closure, Sylow subgroup, supersoluble group, maximal subgroup 20D10, 20D15, 20D30 Let \(G\) be a finite group. Recall that a subgroup \(A\) of \(G\) is said to permute with a subgroup \(B\) if \(AB=BA\). A subgroup \(A\) of \(G\) is said to be \(S\)-quasinormal or \(S\)-permutable in \(G\) if \(A\) permutes with all Sylow subgroups of \(G\). Recall also that \(H^{s G}\) is the \(S\)-permutable closure of \(H\) in \(G\), that is, the intersection of all such \(S\)-permutable subgroups of \(G\) which contain \(H\). We say that \(H\) is Hall \(S\)-permutably embedded in \(G\) if \(H\) is a Hall subgroup of the \(S\)-permutable closure \( H^{s G} \) of \(H\) in \(G\). We prove that the following conditions are equivalent: (1) every subgroup of \(G\) is Hall \(S\)-permutably embedded in \(G\); (2) the nilpotent residual \(G^{\frak{N}}\) of \(G\) is a Hall cyclic of square-free order subgroup of \(G\); (3) \(G = D \rtimes M\) is a split extension of a cyclic subgroup \(D\) of square-free order by a nilpotent group \(M\), where \(M\) and \(D\) are both Hall subgroups of \(G\). Lugansk National Taras Shevchenko University 2017-07-03 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/148 Algebra and Discrete Mathematics; Vol 23, No 2 (2017) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/148/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/148/43 Copyright (c) 2017 Algebra and Discrete Mathematics |
| institution |
Algebra and Discrete Mathematics |
| collection |
OJS |
| language |
English |
| topic |
\(S\)-permutable subgroup Hall \(S\)-permutably embedded subgroup \(S\)-permutable closure Sylow subgroup supersoluble group maximal subgroup 20D10 20D15 20D30 |
| spellingShingle |
\(S\)-permutable subgroup Hall \(S\)-permutably embedded subgroup \(S\)-permutable closure Sylow subgroup supersoluble group maximal subgroup 20D10 20D15 20D30 Sinitsa, Darya A note on Hall S-permutably embedded subgroups of finite groups |
| topic_facet |
\(S\)-permutable subgroup Hall \(S\)-permutably embedded subgroup \(S\)-permutable closure Sylow subgroup supersoluble group maximal subgroup 20D10 20D15 20D30 |
| format |
Article |
| author |
Sinitsa, Darya |
| author_facet |
Sinitsa, Darya |
| author_sort |
Sinitsa, Darya |
| title |
A note on Hall S-permutably embedded subgroups of finite groups |
| title_short |
A note on Hall S-permutably embedded subgroups of finite groups |
| title_full |
A note on Hall S-permutably embedded subgroups of finite groups |
| title_fullStr |
A note on Hall S-permutably embedded subgroups of finite groups |
| title_full_unstemmed |
A note on Hall S-permutably embedded subgroups of finite groups |
| title_sort |
note on hall s-permutably embedded subgroups of finite groups |
| description |
Let \(G\) be a finite group. Recall that a subgroup \(A\) of \(G\) is said to permute with a subgroup \(B\) if \(AB=BA\). A subgroup \(A\) of \(G\) is said to be \(S\)-quasinormal or \(S\)-permutable in \(G\) if \(A\) permutes with all Sylow subgroups of \(G\). Recall also that \(H^{s G}\) is the \(S\)-permutable closure of \(H\) in \(G\), that is, the intersection of all such \(S\)-permutable subgroups of \(G\) which contain \(H\). We say that \(H\) is Hall \(S\)-permutably embedded in \(G\) if \(H\) is a Hall subgroup of the \(S\)-permutable closure \( H^{s G} \) of \(H\) in \(G\). We prove that the following conditions are equivalent: (1) every subgroup of \(G\) is Hall \(S\)-permutably embedded in \(G\); (2) the nilpotent residual \(G^{\frak{N}}\) of \(G\) is a Hall cyclic of square-free order subgroup of \(G\); (3) \(G = D \rtimes M\) is a split extension of a cyclic subgroup \(D\) of square-free order by a nilpotent group \(M\), where \(M\) and \(D\) are both Hall subgroups of \(G\). |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2017 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/148 |
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