A new characterization of finite \(\sigma\)-soluble \(P\sigma T\)-groups
Let \(\sigma =\{\sigma_{i} \mid i\in I\}\) be a partition of the set of all primes \(\mathbb{P}\) and \(G\) a finite group. \(G\) is said to be \emph{\(\sigma\)-soluble} if every chief factor \(H/K\) of \(G\) is a \(\sigma_{i}\)-group for some \(i=i(H/K)\). A set \({\mathcal H}\) of subgroups of \(G...
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| Datum: | 2020 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Lugansk National Taras Shevchenko University
2020
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1530 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543338230972416 |
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| author | Adarchenko, N. M. |
| author_facet | Adarchenko, N. M. |
| author_sort | Adarchenko, N. M. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2020-05-14T18:27:22Z |
| description | Let \(\sigma =\{\sigma_{i} \mid i\in I\}\) be a partition of the set of all primes \(\mathbb{P}\) and \(G\) a finite group. \(G\) is said to be \emph{\(\sigma\)-soluble} if every chief factor \(H/K\) of \(G\) is a \(\sigma_{i}\)-group for some \(i=i(H/K)\). A set \({\mathcal H}\) of subgroups of \(G\) is said to be a complete Hall \(\sigma \)-set of \(G\) if every member \(\ne 1\) of \({\mathcal H}\) is a Hall \(\sigma_{i}\)-subgroup of \(G\) for some \(\sigma_{i}\in \sigma \) and \({\mathcal H}\) contains exactly one Hall \(\sigma_{i}\)-subgroup of \(G\) for every \(i\) such that \(\sigma_{i}\cap \pi (G)\ne \varnothing\). A subgroup \(A\) of \(G\) is said to be \({\sigma}\)-quasinormal or \({\sigma}\)-permutable in \(G\) if \(G\) has a complete Hall \(\sigma\)-set \(\mathcal H\) such that \(AH^{x}=H^{x}A\) for all \(x\in G\) and all \(H\in \mathcal H\). We obtain a new characterization of finite \(\sigma\)-soluble groups \(G\) in which \(\sigma\)-permutability is a transitive relation in \(G\). |
| first_indexed | 2026-02-08T07:57:42Z |
| format | Article |
| id | admjournalluguniveduua-article-1530 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T07:57:42Z |
| publishDate | 2020 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-15302020-05-14T18:27:22Z A new characterization of finite \(\sigma\)-soluble \(P\sigma T\)-groups Adarchenko, N. M. finite group, \(\sigma\)-permutable subgroup, \(P\sigma T\)-group, \(\sigma\)-soluble group, \(\sigma\)-nilpotent group 20D10, 20D15, 20D30 Let \(\sigma =\{\sigma_{i} \mid i\in I\}\) be a partition of the set of all primes \(\mathbb{P}\) and \(G\) a finite group. \(G\) is said to be \emph{\(\sigma\)-soluble} if every chief factor \(H/K\) of \(G\) is a \(\sigma_{i}\)-group for some \(i=i(H/K)\). A set \({\mathcal H}\) of subgroups of \(G\) is said to be a complete Hall \(\sigma \)-set of \(G\) if every member \(\ne 1\) of \({\mathcal H}\) is a Hall \(\sigma_{i}\)-subgroup of \(G\) for some \(\sigma_{i}\in \sigma \) and \({\mathcal H}\) contains exactly one Hall \(\sigma_{i}\)-subgroup of \(G\) for every \(i\) such that \(\sigma_{i}\cap \pi (G)\ne \varnothing\). A subgroup \(A\) of \(G\) is said to be \({\sigma}\)-quasinormal or \({\sigma}\)-permutable in \(G\) if \(G\) has a complete Hall \(\sigma\)-set \(\mathcal H\) such that \(AH^{x}=H^{x}A\) for all \(x\in G\) and all \(H\in \mathcal H\). We obtain a new characterization of finite \(\sigma\)-soluble groups \(G\) in which \(\sigma\)-permutability is a transitive relation in \(G\). Lugansk National Taras Shevchenko University 2020-05-14 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1530 10.12958/adm1530 Algebra and Discrete Mathematics; Vol 29, No 1 (2020) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1530/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/downloadSuppFile/1530/656 Copyright (c) 2020 Algebra and Discrete Mathematics |
| spellingShingle | finite group \(\sigma\)-permutable subgroup \(P\sigma T\)-group \(\sigma\)-soluble group \(\sigma\)-nilpotent group 20D10 20D15 20D30 Adarchenko, N. M. A new characterization of finite \(\sigma\)-soluble \(P\sigma T\)-groups |
| title | A new characterization of finite \(\sigma\)-soluble \(P\sigma T\)-groups |
| title_full | A new characterization of finite \(\sigma\)-soluble \(P\sigma T\)-groups |
| title_fullStr | A new characterization of finite \(\sigma\)-soluble \(P\sigma T\)-groups |
| title_full_unstemmed | A new characterization of finite \(\sigma\)-soluble \(P\sigma T\)-groups |
| title_short | A new characterization of finite \(\sigma\)-soluble \(P\sigma T\)-groups |
| title_sort | new characterization of finite \(\sigma\)-soluble \(p\sigma t\)-groups |
| topic | finite group \(\sigma\)-permutable subgroup \(P\sigma T\)-group \(\sigma\)-soluble group \(\sigma\)-nilpotent group 20D10 20D15 20D30 |
| topic_facet | finite group \(\sigma\)-permutable subgroup \(P\sigma T\)-group \(\sigma\)-soluble group \(\sigma\)-nilpotent group 20D10 20D15 20D30 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1530 |
| work_keys_str_mv | AT adarchenkonm anewcharacterizationoffinitesigmasolublepsigmatgroups AT adarchenkonm newcharacterizationoffinitesigmasolublepsigmatgroups |